Vector Geometry: Worksheets with Answers

Whether you want a homework, some cover work, or a lovely bit of extra practise, this is the place for you. And best of all they all (well, most!) come with answers.

Mathster keyboard_arrow_up Back to Top

Mathster is a fantastic resource for creating online and paper-based assessments and homeworks. They have kindly allowed me to create 3 editable versions of each worksheet, complete with answers.

Corbett Maths keyboard_arrow_up Back to Top

Corbett Maths offers outstanding, original exam style questions on any topic, as well as videos, past papers and 5-a-day. It really is one of the very best websites around.

MATH Worksheets 4 Kids

Child Login

  • Kindergarten
  • Number charts
  • Skip Counting
  • Place Value
  • Number Lines
  • Subtraction
  • Multiplication
  • Word Problems
  • Comparing Numbers
  • Ordering Numbers
  • Odd and Even
  • Prime and Composite
  • Roman Numerals
  • Ordinal Numbers
  • In and Out Boxes
  • Number System Conversions
  • More Number Sense Worksheets
  • Size Comparison
  • Measuring Length
  • Metric Unit Conversion
  • Customary Unit Conversion
  • Temperature
  • More Measurement Worksheets
  • Writing Checks
  • Profit and Loss
  • Simple Interest
  • Compound Interest
  • Tally Marks
  • Mean, Median, Mode, Range
  • Mean Absolute Deviation
  • Stem-and-leaf Plot
  • Box-and-whisker Plot
  • Permutation and Combination
  • Probability
  • Venn Diagram
  • More Statistics Worksheets
  • Shapes - 2D
  • Shapes - 3D
  • Lines, Rays and Line Segments
  • Points, Lines and Planes
  • Transformation
  • Quadrilateral
  • Ordered Pairs
  • Midpoint Formula
  • Distance Formula
  • Parallel, Perpendicular and Intersecting Lines
  • Scale Factor
  • Surface Area
  • Pythagorean Theorem
  • More Geometry Worksheets
  • Converting between Fractions and Decimals
  • Significant Figures
  • Convert between Fractions, Decimals, and Percents
  • Proportions
  • Direct and Inverse Variation
  • Order of Operations
  • Squaring Numbers
  • Square Roots
  • Scientific Notations
  • Speed, Distance, and Time
  • Absolute Value
  • More Pre-Algebra Worksheets
  • Translating Algebraic Phrases
  • Evaluating Algebraic Expressions
  • Simplifying Algebraic Expressions
  • Algebraic Identities
  • Quadratic Equations
  • Systems of Equations
  • Polynomials
  • Inequalities
  • Sequence and Series
  • Complex Numbers
  • More Algebra Worksheets
  • Trigonometry
  • Math Workbooks
  • English Language Arts
  • Summer Review Packets
  • Social Studies
  • Holidays and Events
  • Worksheets >
  • Algebra >

Vector Worksheets

Diverse and comprehensive, our printable vector worksheets successfully cater to high school students. A vector quantity has magnitude and direction. The right way to represent a vector is to draw it as a pointed arrow. The length of the arrow is the magnitude, and the arrowhead denotes the direction of the vector. Learn much more with these pdfs that represent vectors in different forms and notations; practice conversion between different forms; and try your hands at vector operations like adding, subtracting, and multiplying vectors. Try our free vector worksheets and come back for more!

Representing Vectors in Two Ways

Representing Vectors in Two Ways

Gain mastery on expressing vectors in two different forms with these vector pdfs. Instruct high school students to represent each vector in terms of two unit vectors î and ĵ in part A and in <a, b> form in part B.

  • Download the set

Expressing Vectors in Component Form

Expressing Vectors in Component Form

To write the vector in component form, find the horizontal and vertical displacement. In the next part, observe the vectors represented on the graphs and subtract the initial point from the terminal point.

Finding the Magnitude of a Vector

Finding the Magnitude of a Vector

Journey through this collection of vector pdf worksheets to practice finding the magnitude of vectors. Apply the distance formula when the coordinates of the initial point and end point are given.

Finding the Direction Angle of a Vector - Level 1

Finding the Direction Angle of a Vector

Scale up your learning with this bundle of vector worksheets with two levels of difficulty. Use the relevant formula to determine the angle made by the vector with the horizontal axis, the x-axis.

Expressing Vectors in Polar Form

Expressing Vectors in Polar Form

High schoolers represent each vector in the polar form (r,θ) in the first part of these printable vector worksheets, and write the vectors in the Cartesian form in the next part.

Finding Unit Vectors

Finding Unit Vectors

Recalibrate your practice with this amazing set of vector resources on finding vectors with the magnitude of 1. Divide the original vector by its magnitude to find the unit vector that is in the same direction as the given vector.

Adding Two Vectors

Adding Two Vectors

Put your learners in charge as they work their way up in finding the sum of two vectors. Add the corresponding x and y components to find the resultant vector.

Subtracting Two Vectors

Subtracting Two Vectors

Top every test on vector subtraction, addition of a vector with the negative of another vector, with these printable high school worksheets. To find the difference, subtract the corresponding x and y components.

Triangle Method

Resultant of Vectors | Triangle Method and Parallelogram Method

Familiarize yourself with finding the resultant of vectors with these printable vector pdfs. Apply the triangle law of addition/subtraction to calculate the resultant vector in level 1 and use the parallelogram law in level 2.

Multiplying a Vector by a Scalar

Multiplying a Vector by a Scalar

Every high schooler must master vector multiplication! In part A, multiply each vector by a constant (scalar) and write the product. In part B, find the scalar multiple and its magnitude.

Linear Combination of Vectors

Linear Combination of Vectors

Keep the young mathematicians' interest alive with our ready-to-print vector pdf worksheets. Plug in the values in the given vectors, perform the vector operation, and find the linear combination of each vector. More practice in linear combination of vectors!

Related Worksheets

» Trigonometry

» Absolute value

» Inverse Trigonometric Functions

» Quadrants and Angles

Become a Member

Membership Information

Privacy Policy

What's New?

Printing Help

Testimonial

Facebook

Copyright © 2024 - Math Worksheets 4 Kids

This is a members-only feature!

Happy Learning!

Mathwarehouse Logo

Vector Worksheet (pdf)

Resultant vectors sheet with key.

Students will calculate resultant vectors and solve problems involving adding vectors, calculating the magnitude of a resultant as well as the angle formed between two vectors .

Example Questions

Example Question 2.1

Visual Aids

Other details.

This is a 6 part worksheet that includes several model problems plus an answer key

  • Part I Model Problems
  • Part II Vector Basics
  • Part III Addition of Vectors
  • Part IV Find the Magnitude of the Resultant Vector When Two Forces are Applied to an Object
  • Part V Find the Angle Measurements Between the Resultant Vector and Force Vector When Two Forces are Applied to an Object
  • Part VI Answer Key
  • Resultant Vectors
  • Pictures of Vectors
  • Law of Sines and Cosines (pre requisites for many of these problems)

Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there!

Popular pages @ mathwarehouse.com.

Surface area of a Cylinder

geometry vectors worksheet

Visual maths worksheets, each maths worksheet is differentiated and visual.

KS3 and KS4 Vectors Worksheets

Maths Worksheets / KS3 and KS4 Vectors Worksheets

We have curated an outstanding collection of Vectors Worksheets made for students in Years 8 to Year 10. All the worksheets come with answers, and they are aimed at ensuring that children gain a proper understanding of vectors, which is an important mathematical concept for students at KS3 and KS4. We have covered many topics in these worksheets, for example, teaching children about different types of vectors such as unit and column vectors, how to add them together, finding out how big a vector is (called the magnitude) or what position vectors are. They also include cover other important areas, such as parallelism or perpendicularity between vectors, as well as the multiplication of vectors. These are great worksheets for teachers and parents of students in year 8 to year 10 to make learning about this tricky topic enjoyable and engaging!

Column Vectors (A)

Column Vectors Worksheet suitable for students in KS3

Column Vector Addition

Cazoom-Maths.-Vectors.-Column-Vector-Addition

Magnitude of a Vector

Magnitude of a Vector Worksheet perfect for students in year 9 and year 10

Parallel and Perpendicular Vector Search

Parallel and Perpendicular Vector Search Worksheet perfect for students in year 9 and year 10

Position Vectors (A)

Position Vectors Worksheet suitable for students in KS3 and KS4

Position Vectors (B)

Position Vectors Worksheet fit for students in year 9 and year 10

Scalar Multiples

Scalar Multiples Worksheet created for students in KS3 and KS4

Vector Geometry (Word Problems)

Vector Geometry (Word Problems) Worksheet created for students in KS3 and KS4

PRINTABLE PDF VECTORS WORKSHEETS WITH ANSWERS

Check out our downloadable Vectors Worksheets, which will improve your student’s knowledge of various geometrical calculations related to vectors and other related concepts like- Column Vectors, Adding vectors, Two Vectors, Magnitude of a Vector, Position Vectors, and Scalar Multiples. They will also learn about multiplying vectors, vector diagrams, vector geometry and many more concepts. These worksheets are created in easy-to-download PDF format, include answers, and are designed to help your students better understand this complex concept of geometry. These vector worksheets are excellent resources that will make learning fun and exciting, helping your students improve at solving various critical functions related to vectors.

Understanding The Concept Of Vector

Mathematically, vectors are special because they have two fundamental concepts – direction and size. Most of the cases maths only focuses on the amount, but vectors go further than that. We can tell which way to go with a vector rather than just how far. Think of it like this, I can tell you “go 5 mph” or “go 5 mph north”. The second one is much better, isn’t it? In addition to having direction, they also have magnitude. This tells us the strength or amount of the vector’s reach. It’s like adding two forces acting on an object together to find a total effect (which you probably know about from physics class). If you want to strengthen or weaken the force without changing its direction, you can scale up or down a vector. This makes them a powerful tool for modelling and engineering (among other fields). One neat thing about vectors is that they don’t have to stay in one place either; moving parallel to themselves anywhere in space still represents the same thing.

Properties Of Vector

The unique thing about vectors in mathematics is that they have two properties, magnitude and direction. While many mathematical ideas only deal with numerical value, vectors are different since they also show where an object is going. For instance, there is a difference between “In 10 steps” and “10 steps to the east”. This one, like a vector, gives more information as it tells both direction and magnitude of the distance. The other use of the word magnitude refers to how large or small something is. Imagine that a vector can be represented by a stretchy arrow with a certain length indicating its magnitude and its tip; pointing towards the destination would depict their directions. What’s great about vectors is that they need not be fixed in one location either; anywhere parallel move represents them all in space.

Using Vector In Real Life

Although vectors are a crucial topic in mathematics and geometry, they are basic tools used to represent quantities that have both magnitude and direction. Several ways of using vectors in our everyday lives exist. For instance, the routes of planes from one city to another represent one of the most obvious examples of these applications. With a vector approach, it is possible to give a precise description of its course and speed. In sports, an interesting issue could be about the direction as well as force applied when kicking a football. We can answer this by using vectors to indicate where the ball will go towards the goal’s midpoint and how fast it will get there. Vectoring also has great significance in animation because it helps us determine how objects move or interact with each other. When crossing waters or air travel, vectors become important in calculating courses because they consider factors like wind speeds or directions for instance. In other words, determining where something is going isn’t just about knowing how far.

Get 20 FREE MATHS WORKSHEETS

Fill out the form below to get 20 FREE maths worksheets!

One to one maths interventions built for KS4 success

Weekly online one to one GCSE maths revision lessons now available

In order to access this I need to be confident with:

Ratio and proportion

Properties of 2D shapes

This topic is relevant for:

GCSE Maths

Vector problems

Vector Problems

Here we will learn about more difficult vector problems, including vector routes involving midpoints, fractions and ratios of lengths. We will also look at parallel vectors.

There are also vector worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What are vector problems?

Vector problems use vectors to solve a variety of different types of problems. Vectors have both a magnitude and direction and can be used to show a movement. A quantity which has just magnitude (size) is called a scalar.

For example,

We can write vectors in several ways,

  • Using an arrow
  • Using boldface  

What are vector problems?

Key facts for vector problems

In order to solve problems involving vectors it is helpful to use several key facts.

geometry vectors worksheet

How to solve vector problems

In order to solve vector problems:

Write any information you know onto the diagram.

Decide the route.

Write the vector.

Simplify your answer.

Explain how to solve vector problems

Explain how to solve vector problems

Vector problems worksheet

Get your free vector problems worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Related lessons on vectors

Vector problems is part of our series of lessons to support revision on vectors . You may find it helpful to start with the main vectors lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

  • Magnitude of a vector  
  • Column vector
  • Vector notation
  • Vector multiplication
  • Vector addition
  • Vector subtraction

Vector problem examples

Example 1: parallel lines.

The shape below is made from 8 equilateral triangles.

Find the vector \overrightarrow{QT} .

Parallel vectors of the same magnitude are the same. These triangles are all identical therefore we can label the corresponding vectors a and b .

2 Decide the route.

We need to find a route where we know the vectors.

3 Write the vector.

4 Simplify your answer.

Example 2: extended line

The line AB is extended to the point D so that the length AD is three times the length AB .

Find the vector \overrightarrow{AD} .

There is currently no further information we can add to the diagram.

We know AD is three times the length of AB . We need to find a route from A to B and then multiply it by three.

Example 3: midpoint

D is the midpoint of AC . Find the vector \overrightarrow{BD} .

When we are given information about midpoints, fractions of lines or ratios, it can be helpful to add this information onto the diagram. Here we are going to use the fraction  \frac{1}{2} to show that D is half way along the line.

We are trying to find a route from B to D . We know that \overrightarrow{BC}=\textbf{p} and we then need to get from C to D . We also know that CD=\frac{1}{2}CA .

We can find the vector \overrightarrow{CA} and then half it.

\begin{aligned} &\overrightarrow{CA}=-\textbf{p}+\textbf{q}\\\\ &\overrightarrow{CD}=\frac{1}{2}\overrightarrow{CA}=\frac{1}{2}(-\textbf{p}+\textbf{q}) \end{aligned}

Now we have a route from B to D .

Example 4: fraction of a line

Find the vector \overrightarrow{MN} .

When we are given information about midpoints, fractions of lines or ratios, it can be helpful to add this information onto the diagram. Here we know

\text{BM }=\frac{1}{4}\text{ BA}, \text{ AN} = \frac{1}{3} \text{ AC} .

We need to find a route from M to N . We can see that

\text{MA}=\frac{3}{4}\text{BA and AN}=\frac{1}{3}\text{AC} .

\overrightarrow{MA}=\frac{3}{4}\overrightarrow{BA}=\frac{3}{4}(-2\textbf{a})=-\frac{3}{2}\textbf{a}\\\overrightarrow{AN}=\frac{1}{3}\overrightarrow{AC}=\frac{1}{3}(6\textbf{b})=2\textbf{b}

We now have a route from M to N .

The answer here cannot be simplified.

Example 5: ratio

ED =2EH and the point J is such that GJ:JH = 2:1 .

Find the vector \overrightarrow{JD} .

When we are given information about midpoints, fractions of lines or ratios, it can be helpful to add this information onto the diagram. Here we know GJ:JH=2:1 . This means that  \text{GJ}=\frac{2}{3}\text{GH and JH}=\frac{1}{3}\text{GH} .

We also know that ED=2EH therefore HD is the same length as EH and in the same direction.

We know  \text{JH}=\frac{1}{3}\text{GH} .

We can find the vector  \overrightarrow{GH} .

\begin{aligned} &\overrightarrow{GH}=-\textbf{a}-\textbf{b}+2\textbf{a}=\textbf{a}-\textbf{b}\\\\ &\overrightarrow{JH}=\frac{1}{3}\overrightarrow{GH}=\frac{1}{3}(\textbf{a}-\textbf{b}) \end{aligned}

We now have a route from J to D .

Example 6: mix of information

E is the point on AB such that \text{EB }=\frac{1}{3}\text{ AB} .

D is the point on BC such that CD:DB=1:4 .

F is the midpoint of ED .

Find the vector \overrightarrow{EF} .

We want to find the the vector \overrightarrow{EF} .

We know that \text{EF}=\frac{1}{2}\text{ED} so we can start by finding the vector \overrightarrow{ED} .

\begin{aligned} &\overrightarrow{EB}=\frac{1}{3}\overrightarrow{AB}=\frac{1}{3}(6\textbf{a})=2\textbf{a}\\\\ &\overrightarrow{BD}=\frac{4}{5}\overrightarrow{BC}=\frac{4}{5}(10\textbf{b})=8\textbf{b} \end{aligned}

\overrightarrow{ED}=2\textbf{a}+8\textbf{b}

This answer cannot be simplified.

How to show two vectors are parallel

Two vectors are parallel if one is a multiple of the other. This is because if one vector is a multiple of another, it is a bigger or smaller version of the other.

In order to show two vectors are parallel:

Work out each vector.

Show that one is a multiple of the other by factorising.

Showing two vectors are parallel examples

Example 7: two lines are parallel.

Show that AB is parallel to CD .

We need to find the vector \overrightarrow{AB} .

\begin{aligned} &\overrightarrow{AB}=4\textbf{a}+\textbf{a}+2\textbf{b}-3\textbf{a}+2\textbf{b}\\\\ &\overrightarrow{AB}=2\textbf{a}+4\textbf{b} \end{aligned}

Since the vector  \overrightarrow{AB} is a multiple of the vector  \overrightarrow{CD} these vectors are parallel.

Example 8: two line segments form one straight line

ABCD is a parallelogram.

The line AB is extended to the point E such that \text{BE}=\frac{2}{3}\text{AB} .

The point F is on the line BC such that BF:FC=2:3 .

Show that DFE is a straight line.

We need to find the vectors \overrightarrow{DF} \text{ and } \overrightarrow{FE} .

\begin{aligned} &\overrightarrow{CF}=\frac{3}{5}\overrightarrow{CB}=3\textbf{b}\\\\ &\overrightarrow{FB}=\frac{2}{5}\overrightarrow{CB}=2\textbf{b} \end{aligned}

\begin{aligned} &\overrightarrow{DF}=6\textbf{a}+3\textbf{b}\\\\ &\overrightarrow{FE}=4\textbf{a}+2\textbf{b} \end{aligned}

Since \overrightarrow{DF} is a multiple of \overrightarrow{FE} the vectors are parallel. They also share the point F meaning they form a straight line.

Common misconceptions

  • Using the wrong sign

Remember to make the vector negative when going backwards along it.

  • Mistakes with ratios

If two parts of a line are in the ratio 1:4, this means one part is ⅕ of a line and the other part ⅘ .

Practice vector problem questions

1. ABCD is a parallelogram. The line AD is extended to the point E so that AE=3AD .

Find the vector  \overrightarrow{CE} .

GCSE Quiz False

2. \overrightarrow{PR}=4\textbf{a}, \overrightarrow{PQ}=3\textbf{b}

M is the midpoint of QR .

Find the vector \overrightarrow{RM} .

3. \overrightarrow{AB}=3\textbf{b}, \overrightarrow{BC}=2\textbf{a}

AD=2BC and the point E is on the line CD such that \text{ED}=\frac{1}{3}\text{CD} .

Find the vector \overrightarrow{AE} .

We know \text{DC}=\frac{1}{3}\text{DC} so we need to find the vector \overrightarrow{DC} .

4. \overrightarrow{AC}=9\textbf{a}, \overrightarrow{AB}=8\textbf{b}

The point N is such that \text{AN}=\frac{1}{3}\text{AC} , and the point P is such that AP:PB=1:3 . M is the midpoint of BN . Find the vector \overrightarrow{PM} .

We know that BM=\frac{1}{2}BN so we need to find the vector \overrightarrow{BN} .

5. \overrightarrow{AB}=2\textbf{a}+6\textbf{b}, \overrightarrow{AC}=3\textbf{a}+2\textbf{b}

D is the midpoint of BC and E is the point on AD such that AE:ED=1:3 .

We need to begin by finding the vector \overrightarrow{AD} .

6. \overrightarrow{AB}=4\textbf{a}, \overrightarrow{AE}=8\textbf{b} D is the midpoint of AE .

4\textbf{b}-4\textbf{a}=4(\textbf{b}-\textbf{a}) so \overrightarrow{CE} is parallel to \textbf{b}-\textbf{a} .

Vector problems GCSE questions

1. \overrightarrow{AB}=6\textbf{b},\overrightarrow{BC}=9\textbf{a},\overrightarrow{DF}=-3\textbf{a}+3\textbf{b}

\overrightarrow{BC} =3\overrightarrow{AD} – they are multiples of each other therefore parallel.

2. \overrightarrow{AD}=12\textbf{a}, \overrightarrow{CD}=8\textbf{b}

E is the midpoint of AB .

3. \overrightarrow{AB}=4\textbf{a}, \overrightarrow{BC}=2\textbf{b}

D is the midpoint of AB.

Point E is such that AE:EC=3:1 .

Determine whether DEF is a straight line.

DEF is not a straight line since \overrightarrow{DE} and \overrightarrow{EF} and are not multiples of each other and therefore are not parallel.

Learning checklist

You have now learned how to:

  • Solve complex problems involving vectors

The next lessons are

  • Loci and construction
  • Transformations
  • Circle theorems

Still stuck?

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

GCSE Benefits

Find out more about our GCSE maths tuition programme.

Privacy Overview

Math Worksheets Land

Math Worksheets For All Ages

  • Math Topics
  • Grade Levels

Vector Worksheets

What are Vectors? Vectors is a term in science that denotes anything that has a magnitude as well as a direction. The right way to represent a vector is to draw it as a pointed arrow. The length of the arrow is the magnitude and the arrowhead points in the direction. The most basic approach to add two vectors is to join one's head to the tail of the other vector. The addition of two vectors obeys the commutative property which means that no matter in which order you add the vectors, the result will be the same. We use the Pythagora's theorem to calculate the magnitude of a vector. A vector that has a magnitude 1 is called a unit vector. When you multiply two vectors, you always get a vector. The product of two vectors is called a cross product. The order in which you multiply two vectors is very important as; A × B ≠ B × A. The application of vectors in today's world is endless. They can help understand and chart very complex systems.

  • Adding Vectors End to End - Students learn a fundamental way to learn more about a system.
  • Drawing Vectors - We visualize a way to better understand the nature of vectors and get a full sense of their force and direction.
  • Finding the Components of a Vector - We show students how to determine both the x and y components of a vector.
  • Magnitudes of Scalar Multiples - We look at the methods we can use to measure these distances.
  • Multiplying a Vector by a Matrix - We show students how perform this will column and row vectors.
  • Multiply a Vector by a Scalar - We look at how we breakdown the vector into components and then multiply each by the same scalar.
  • Vector Based Word Problems - These types of problems have a great many applications in the field of physics.
  • Vector Subtraction - This helps us understand the differences between two vectors.
  • Vector Sums Magnitude and Direction - We look at what these are and what they mean in different situations and how to make sense of them.

Get Access to Answers, Tests, and Worksheets

Become a paid member and get:

  • Answer keys to everything
  • Unlimited access - All Grades
  • 64,000 printable Common Core worksheets, quizzes, and tests
  • Used by 1000s of teachers!

Worksheets By Email:

Get Our Free Email Now!

We send out a monthly email of all our new free worksheets. Just tell us your email above. We hate spam! We will never sell or rent your email.

Thanks and Don't Forget To Tell Your Friends!

I would appreciate everyone letting me know if you find any errors. I'm getting a little older these days and my eyes are going. Please contact me, to let me know. I'll fix it ASAP.

  • Privacy Policy
  • Other Education Resource

© MathWorksheetsLand.com, All Rights Reserved

Library homepage

  • school Campus Bookshelves
  • menu_book Bookshelves
  • perm_media Learning Objects
  • login Login
  • how_to_reg Request Instructor Account
  • hub Instructor Commons
  • Download Page (PDF)
  • Download Full Book (PDF)
  • Periodic Table
  • Physics Constants
  • Scientific Calculator
  • Reference & Cite
  • Tools expand_more
  • Readability

selected template will load here

This action is not available.

Mathematics LibreTexts

11.3E: Exercises for The Dot Product

  • Last updated
  • Save as PDF
  • Page ID 8631

For exercises 1-4, the vectors \(\vecs{u}\) and \(\vecs{v}\) are given. Calculate the dot product \(\vecs{u}\cdot\vecs{v}\).

1) \(\quad \vecs{u}=⟨3,0⟩, \quad \vecs{v}=⟨2,2⟩\)

2) \(\quad \vecs{u}=⟨3,−4⟩, \quad \vecs{v}=⟨4,3⟩\)

3) \(\quad \vecs{u}=⟨2,2,−1⟩, \quad \vecs{v}=⟨−1,2,2⟩\)

4) \(\quad \vecs{u}=⟨4,5,−6⟩, \quad \vecs{v}=⟨0,−2,−3⟩\)

For exercises 5-8, the vectors \(\vecs{a}, \,\vecs{b}\), and \(\vecs{c}\) are given. Determine the vectors \((\vecs{a}\cdot\vecs{b})\vecs{c}\) and \((\vecs{a}⋅\vecs{c})\vecs{b}.\) Express the vectors in component form.

5) \(\quad \vecs{a}=⟨2,0,−3⟩, \quad \vecs{b}=⟨−4,−7,1⟩, \quad \vecs{c}=⟨1,1,−1⟩\)

6) \(\quad \vecs{a}=⟨0,1,2⟩, \quad \vecs{b}=⟨−1,0,1⟩, \quad \vecs{c}=⟨1,0,−1⟩\)

7) \(\quad \vecs{a}=\mathbf{\hat i} +\mathbf{\hat j} , \quad \vecs{b}=\mathbf{\hat i} −\mathbf{\hat k} , \quad \vecs{c}=\mathbf{\hat i} −2\mathbf{\hat k} \)

8) \(\quad \vecs{a}=\mathbf{\hat i} −\mathbf{\hat j} +\mathbf{\hat k} , \quad \vecs{b}=\mathbf{\hat j} +3\mathbf{\hat k} , \quad \vecs{c}=−\mathbf{\hat i} +2\mathbf{\hat j} −4\mathbf{\hat k} \)

For exercises 9-12, two vectors are given.

a. Find the measure of the angle \(θ\) between these two vectors. Express the answer in radians rounded to two decimal places, if it is not possible to express it exactly.

b. Is \(θ\) an acute angle?

9) [T] \(\quad \vecs{a}=⟨3,−1⟩, \quad \vecs{b}=⟨−4,0⟩\)

10) [T] \(\quad \vecs{a}=⟨2,1⟩, \quad \vecs{b}=⟨−1,3⟩\)

11) \(\quad \vecs{u}=3\mathbf{\hat i}, \quad \vecs{v}=4\mathbf{\hat i} +4\mathbf{\hat j} \)

12) \(\quad \vecs{u}=5\mathbf{\hat i}, \quad \vecs{v}=−6\mathbf{\hat i} +6\mathbf{\hat j} \)

For exercises 13-18, find the measure of the angle between the three-dimensional vectors \(\vecs{a}\) and \(\vecs{b}\). Express the answer in radians rounded to two decimal places, if it is not possible to express it exactly.

13) \(\quad \vecs{a}=⟨3,−1,2⟩, \quad \vecs{b}=⟨1,−1,−2⟩\)

14) \(\quad \vecs{a}=⟨0,−1,−3⟩, \quad \vecs{b}=⟨2,3,−1⟩\)

15) \(\quad \vecs{a}=\mathbf{\hat i} +\mathbf{\hat j} , \quad \vecs{b}=\mathbf{\hat j} −\mathbf{\hat k} \)

16) \(\quad \vecs{a}=\mathbf{\hat i} −2\mathbf{\hat j} +\mathbf{\hat k} , \quad \vecs{b}=\mathbf{\hat i} +\mathbf{\hat j} −2\mathbf{\hat k} \)

17) [T] \(\quad \vecs{a}=3\mathbf{\hat i} −\mathbf{\hat j} −2\mathbf{\hat k} , \quad \vecs{b}=\vecs v+\vecs w,\) where \(\quad \vecs{v}=−2\mathbf{\hat i} −3\mathbf{\hat j} +2\mathbf{\hat k} \) and \(\vecs{w}=\mathbf{\hat i} +2\mathbf{\hat k} \)

18) [T] \(\quad \vecs{a}=3\mathbf{\hat i} −\mathbf{\hat j} +2\mathbf{\hat k} , \quad \vecs{b}=\vecs v−\vecs w,\) where \(\quad \vecs{v}=2\mathbf{\hat i} +\mathbf{\hat j} +4\mathbf{\hat k} \) and \(\vecs{w}=6\mathbf{\hat i} +\mathbf{\hat j} +2\mathbf{\hat k} \)

For exercises 19-22, determine whether the given vectors are orthogonal.

19) \(\quad \vecs{a}=⟨x,y⟩, \quad \vecs{b}=⟨−y, x⟩\), where \(x\) and \(y\) are nonzero real numbers

20) \(\quad \vecs{a}=⟨x, x⟩, \quad \vecs{b}=⟨−y, y⟩\), where \(x\) and \(y\) are nonzero real numbers

21) \(\quad \vecs{a}=3\mathbf{\hat i} −\mathbf{\hat j} −2\mathbf{\hat k} , \quad \vecs{b}=−2\mathbf{\hat i} −3\mathbf{\hat j} +\mathbf{\hat k} \)

22) \(\quad \vecs{a}=\mathbf{\hat i} −\mathbf{\hat j} , \quad \vecs{b}=7\mathbf{\hat i} +2\mathbf{\hat j} −\mathbf{\hat k} \)

23) Find all two-dimensional vectors \(\vecs a\) orthogonal to vector \( \vecs b=⟨3,4⟩.\) Express the answer in component form.

24) Find all two-dimensional vectors \( \vecs a\) orthogonal to vector \( \vecs b=⟨5,−6⟩.\) Express the answer by using standard unit vectors.

25) Determine all three-dimensional vectors \( \vecs u\) orthogonal to vector \( \vecs v=⟨1,1,0⟩.\) Express the answer by using standard unit vectors.

26) Determine all three-dimensional vectors \(\vecs u\) orthogonal to vector \(\vecs v=\mathbf{\hat i} −\mathbf{\hat j} −\mathbf{\hat k} \). Express the answer in component form.

27) Determine the real number \(α\) such that vectors \(\vecs a=2\mathbf{\hat i} +3\mathbf{\hat j} \) and \(\vecs b=9\mathbf{\hat i} +α\mathbf{\hat j} \) are orthogonal.

28) Determine the real number \(α\) such that vectors \(\vecs a=−3\mathbf{\hat i} +2\mathbf{\hat j} \) and \(\vecs b=2\mathbf{\hat i} +α\mathbf{\hat j} \) are orthogonal.

29) [T] Consider the points \(P(4,5)\) and \(Q(5,−7)\), and note that \(O\) represents the origin.

a. Determine vectors \(\vecd{OP}\) and \(\vecd{OQ}\). Express the answer by using standard unit vectors.

b. Determine the measure of angle \(O\) in triangle \(OPQ\). Express the answer in degrees rounded to two decimal places.

30) [T] Consider points \( A(1,1), B(2,−7),\) and \(C(6,3)\).

a. Determine vectors \( \vecd{BA}\) and \(\vecd{BC}\). Express the answer in component form.

b. Determine the measure of angle \(B\) in triangle \(ABC\). Express the answer in degrees rounded to two decimal places.

31) Determine the measure of angle \(A\) in triangle \(ABC\), where \(A(1,1,8), B(4,−3,−4),\) and \(C(−3,1,5).\) Express your answer in degrees rounded to two decimal places.

32) Consider points \(P(3,7,−2)\) and \(Q(1,1,−3).\) Determine the angle between vectors \(\vecd{OP}\) and \(\vecd{OQ}\). [Remember that \(O\) represents the origin.] Express the answer in degrees rounded to two decimal places.

For exercises 33-34, determine which (if any) pairs of the following vectors are orthogonal.

33) \(\quad\vecs u=⟨3,7,−2⟩, \quad \vecs v=⟨5,−3,−3⟩, \quad \vecs w=⟨0,1,−1⟩\)

34) \(\quad\vecs u=\mathbf{\hat i} −\mathbf{\hat k} , \quad \vecs v=5\mathbf{\hat j} −5\mathbf{\hat k} , \quad \vecs w=10\mathbf{\hat j} \)

35) Use vectors to show that a parallelogram with equal diagonals is a rectangle .

36) Use vectors to show that the diagonals of a rhombus are perpendicular.

37) Show that \(\vecs u⋅(\vecs v+\vecs w)=\vecs u⋅\vecs v+\vecs u⋅\vecs w\) is true for any vectors \(\vecs u, \vecs v\), and \(\vecs w\).

38) Verify the identity \(\vecs u⋅(\vecs v+\vecs w)=\vecs u⋅\vecs v+\vecs u⋅\vecs w\) for vectors \(\vecs u=⟨1,0,4⟩, \vecs v=⟨−2,3,5⟩,\) and \(\vecs w=⟨4,−2,6⟩\).

For exercises 39-41, determine \(\vecs u\cdot\vecs v\) using the given information.

39) \(\quad\|\vecs u\| = 5\), \(\quad\|\vecs v\| = 3\), and the angle between \(\vecs u\) and \(\vecs v\) is \(\pi/6\) rad.

40) \(\quad\|\vecs u\| = 20\), \(\quad\|\vecs v\| = 15\), and the angle between \(\vecs u\) and \(\vecs v\) is \(5\pi/4\) rad.

41) \(\quad\|\vecs u\| = 7\), \(\quad\|\vecs v\| = 12\), and the angle between \(\vecs u\) and \(\vecs v\) is \(\pi/2\) rad.

42) Considering the definition of the dot product, \(\vecs u\cdot\vecs v = \|\vecs u\|\|\vecs v\|\cos \theta\), where \(\theta\) is the angle between \(\vecs u\) and \(\vecs v\), what can you say about the angle \(\theta\) between two nonzero vectors \(\vecs u\) and \(\vecs v\) if:

a. \(\quad\vecs u \cdot \vecs v > 0\)?

b. \(\quad\vecs u \cdot \vecs v < 0\)?

c. \(\quad\vecs u \cdot \vecs v = 0\)?

In exercises 43-45, you are given the vertices of a triangle. Use dot products to determine whether each triangle is acute , obtuse , or right .

43) \(\quad (2, 3, 0), \, (3, 1, -2), \, (-1, 4, 5)\)

44) \(\quad (5, 1, 0), \, (7, 1, 1), \, (6, 3, 2)\)

45) \(\quad (6, 1, 4), \, (3, 2, -1), \, (2, 3, 1)\)

46) Which of the following operations are allowed for nonzero vectors \(\vecs u\), \(\vecs v\), and \(\vecs w\), and which are not. Explain your reasoning.

a. \(\vecs u + \left(\vecs v \cdot \vecs w\right)\) b. \(\left(\vecs u \cdot \vecs v\right) \cdot \vecs w\) c. \(\left(\vecs u \cdot \vecs v\right) \vecs w\) d. \(\left(\vecs u + \vecs v\right) \cdot \vecs w\) e. \(\left(\vecs u + \vecs v\right) \cdot \|\vecs w\|\) f. \(\|\vecs u + \vecs v\| \|\vecs w\|\)

Projections

For the exercises 47-50, given the vectors \(\vecs u\) and \(\vecs v\):

a. Find the vector projection \(\text{Proj}_\vecs{u}\vecs v\) of vector \(\vecs v\) onto vector \(\vecs u\) and the component of \(\vecs v\) that is orthogonal to \(\vecs u\), i.e., \(\vecs v_\text{perp}\). Express your answers in component form.

b. Find the scalar projection \(\text{comp}_\vecs{u}\vecs v = \| \text{Proj}_\vecs{u}\vecs v \|\) of vector \(\vecs v\) onto vector \(\vecs u\).

c. Find the vector projection \(\text{Proj}_\vecs{v}\vecs u\) of vector \(\vecs u\) onto vector \(\vecs v\) and the component of \(\vecs u\) that is orthogonal to \(\vecs v\), i.e., \(\vecs u_\text{perp}\). Express your answers in unit vector form.

d. Find the scalar projection \(\text{comp}_\vecs{v}\vecs u\) of vector \(\vecs u\) onto vector \(\vecs v\).

47) \(\quad\vecs u=5\mathbf{\hat i} +2\mathbf{\hat j} , \quad \vecs v=2\mathbf{\hat i} +3\mathbf{\hat j} \)

48) \(\quad\vecs u=⟨−4,7⟩,\quad \vecs v=⟨3,5⟩\)

49) \(\quad\vecs u=3\mathbf{\hat i} +2\mathbf{\hat k} , \quad \vecs v=2\mathbf{\hat j} +4\mathbf{\hat k} \)

50) \(\quad\vecs u=⟨4,4,0⟩, \quad \vecs v=⟨0,4,1⟩\)

51) Consider the vectors \(\vecs u=4\mathbf{\hat i} −3\mathbf{\hat j} \) and \(\vecs v=3\mathbf{\hat i} +2\mathbf{\hat j} .\)

a. Find the component form of vector \(\text{Proj}_\vecs{u}\vecs v\) that represents the projection of \(\vecs v\) onto \(\vecs u\).

b. Write the decomposition \(\vecs v=\vecs w+\vecs q\) of vector \(\vecs v\) into the orthogonal components \(\vecs w\) and \(\vecs q\), where \(\vecs w\) is the projection of \(\vecs v\) onto \(\vecs u\) and \(\vecs q\) is the vector component of \(\vecs v\) orthogonal to the direction of \(\vecs u\). That is, \( \vecs q = \vecs v_\text{perp}\).

52) Consider vectors \(\vecs u=2\mathbf{\hat i} +4\mathbf{\hat j} \) and \(\vecs v=4\mathbf{\hat j} +2\mathbf{\hat k} .\)

a. Find the component form of vector \(\vecs w=\text{Proj}_\vecs{u}\vecs v\) that represents the projection of \(\vecs v\) onto \(\vecs u\).

b. Write the decomposition \(\vecs v=\vecs w+\vecs q\) of vector \(\vecs v\) into the orthogonal components \(\vecs w\) and \(\vecs q\), where \(\vecs w\) is the projection of \(\vecs v\) onto \(\vecs u\) and \(\vecs q\) is a vector orthogonal to the direction of \(\vecs u\).

53) A \(50,000\)-pound truck is parked on a hill with a \(5°\) slope above the horizontal (in the positive \(x\)-direction). Considering only the force due to gravity, find

a. the component of the weight of the truck that is pulling the truck down the hill (aligned with the road)

b. the component of the weight of the truck that is perpendicular to the road.

c. the magnitude of the force needed to keep the truck from rolling down the hill

d. the magnitude of the force that is perpendicular to the road.

54) Given the vectors \(\vecs u\) and \(\vecs v\) shown in each diagram below, draw in \(\text{Proj}_\vecs{v}\vecs u\) and \( \vecs u_\text{perp} \).

acuteAngle.png

55) Find the work done by force \(\vecs F=⟨5,6,−2⟩\) (measured in Newtons) that moves a particle from point \(P(3,−1,0)\) to point \(Q(2,3,1)\) along a straight line (the distance is measured in meters).

56) [T] A sled is pulled by exerting a force of 100 N on a rope that makes an angle of \(25°\) with the horizontal. Find the work done in pulling the sled 40 m. (Round the answer to one decimal place.)

57) [T] A father is pulling his son on a sled at an angle of \(20°\) with the horizontal with a force of 25 lb (see the following image). He pulls the sled in a straight path of 50 ft. How much work was done by the man pulling the sled? (Round the answer to the nearest integer.)

CNX_Calc_Figure_12_03_204.jfif

58) [T] A car is towed using a force of 1600 N. The rope used to pull the car makes an angle of 25° with the horizontal. Find the work done in towing the car 2 km. Express the answer in joules (\(1\) J\(=1\) N⋅m) rounded to the nearest integer.

59) [T] A boat sails north aided by a wind blowing in a direction of \( N30°E\) with a magnitude of 500 lb. How much work is performed by the wind as the boat moves 100 ft? (Round the answer to two decimal places.)

Other Applications of the Dot Product

60) [T] Find the vectors that join the center of a clock to the hours 1:00, 2:00, and 3:00. Assume the clock is circular with a radius of 1 unit.

61) A methane molecule has a carbon atom situated at the origin and four hydrogen atoms located at points \(P(1,1,−1),Q(1,−1,1),R(−1,1,1),\) and \(S(−1,−1,−1)\) (see figure).

a. Find the distance between the hydrogen atoms located at \(P\)and \(R\).

b. Find the angle between vectors \(\vecd{OS}\) and \(\vecd{OR}\) that connect the carbon atom with the hydrogen atoms located at \(S\) and \(R\), which is also called the bond angle . Express the answer in degrees rounded to two decimal places.

CNX_Calc_Figure_12_03_012.jfif

62) Vector \(\vecs p=⟨150,225,375⟩\) represents the price of certain models of bicycles sold by a bicycle shop. Vector \(\vecs n=⟨10,7,9⟩\) represents the number of bicycles sold of each model, respectively. Compute the dot product \(\vecs p⋅\vecs n\) and state its meaning.

63) [T] Two forces \(\vecs F_1\) and \(\vecs F_2\) are represented by vectors with initial points that are at the origin. The first force has a magnitude of 20 lb and passes through the point \(P(1,1,0)\). The second force has a magnitude of 40 lb and passes through the point \(Q(0,1,1)\). Let \(\vecs F\) be the resultant force of forces \(\vecs F_1\) and \(\vecs F_2\).

a. Find the magnitude of \(\vecs F\). (Round the answer to one decimal place.)

b. Find the direction angles of \(\vecs F\). (Express the answer in degrees rounded to one decimal place.)

64) [T] Consider \(\vecs r(t)=⟨\cos t,\sin t,2t⟩\) the position vector of a particle at time \(t∈[0,30]\), where the components of \(\vecs r\) are expressed in centimeters and time in seconds. Let \(\vecd{OP}\) be the position vector of the particle after 1 sec.

a. Show that all vectors \(\vecd{PQ}\), where \(Q(x,y,z)\) is an arbitrary point, orthogonal to the instantaneous velocity vector \(\vecs v(1)\) of the particle after 1 sec, can be expressed as \( \vecd{PQ}=⟨x−\cos 1,y−\sin 1,z−2⟩\), where \(x\sin 1−y\cos 1−2z+4=0.\) The set of point \(Q\) describes a plane called the normal plane to the path of the particle at point \(P\).

b. Use a CAS to visualize the instantaneous velocity vector and the normal plane at point \(P\) along with the path of the particle.

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org .

Exercises and LaTeX edited by Paul Seeburger. Problems 39-46 and 53 & 54 were created by Paul Seeburger. Solution to Problem 59 also was added by Paul Seeburger.

Vector Geometry

In these lessons, we will look at some examples of problems involving vectors in geometrical shapes.

Related Pages Vectors Equal Vectors Negative Vectors Vector Multiplication

vector geometry

How to solve vector geometry problems? GCSE Revision Questions on Vectors Topics in this lesson: Vectors with numbers, magnitude of a vector, algebraic vectors, parallel vectors

ABCDEF is a regular hexagon. AB = n a) Explain why ED = n. BC = m, CD = p b) Find (i) AC, (ii) AD c) What is FD?

ABC is a straight line where BC = 3AB. OA = a, AB = b Express OC in terms of a and b.

In triangle OAB, OA = a and OB = b. (i) Find in terms of a and b, the vector AB. P is the midpoint of AB. (ii) Find in terms of a and b, the vector AP. (iii) Find in terms of a and b, the vector OP.

OABC is a parallelogram with OA = a and OB = b. E is a point on AC such that AE = 1/4 AC F is a point on BC such that BF = 1/4 BC. (a) Find in terms of a and b (i) AB, (ii) AE, (iii) OE, (iv) OF, (v) EF (b) Write down two geometric properties connecting EF and AB

Exam Question 1: AB = 2x and BC = 4y, ABCD is a straight line. (a) Write down the vector AC in terms of x and y. (b) AC:CD = 2:1 Works out the vector AD in terms of x and y. Give your answer as simply as possible.

Exam Question 2: WXYZ is a trapezium WX = s, WZ = t, ZY:WX = 3:2 (a) Write vector ZY in terms of s (b) Work out vector XY in terms of s and t. Give your answer in its simplest form.

Exam Question 3: PQRS is a trapezium as shown (a) Write down in terms of a and b vector SR. (b) Work out in terms of a and b vector QR. Give your answer as simply as possible.

How to do Vectors? A/A* GCSE Maths revision Higher level worked exam questions (include straight lines) GCSE Higher maths vector geometry questions including proof of straight and parallel lines grades A/A*. Vectors on trapeziums, Vectors on triangles, Vectors on rectangles and parallelograms

Geometric Vectors Part 1 This video introduces Geometric Vectors, along with the magnitude, opposite vectors, congruent vectors, and resultants.

  • A vector is a quantity that has both magnitude and direction. (It looks like a directed line segment).
  • The length of a line segment is the magnitude. The direction indicates the direction of the vector.
  • A vector with its initial point at the origin is in standard position.
  • The direction of the vector is directed angle between the positive x-axis and the vector.
  • If both the initial point and the terminal point are at the origin, it is called a zero vector.

Geometric Vectors with Application Problems In a rowing exercise, John was rowing directly across a river at the rate of 4 mph. The current was flowing at a rate of 3 mph. Use a ruler to draw each vector to scale and draw a vector to represent the path of the boat. Determine the magnitude of the resultant velocity of the boat by measuring the vector.

A ship leaving port sails for 25 miles in a direction 35° north of due east. Find the magnitude of the vertical and horizontal component.

A piling for a high-rise building is pushed by two bulldozers at exactly the same time. One bulldozer exert a force of 1550 pounds in a westerly direction. The other bulldozer pushes the piling with a force of 3050 pounds in a northerly direction. a. Find the magnitude of the resultant force upon the piling, to the nearest pound. b. What is the directions of the resultant force upon the piling, to the nearest pound.

Geometric Proofs using Vectors

  • Prove the diagonals of a parallelogram meet at right angles if and only if it is a rhombus.
  • Prove the midpoints of the sides of a quadrilateral join to form a parallelogram.
  • Prove the sum of the squares of the lengths of a parallelogram’s diagonals is equal to the sum of the squares of the lengths of the sides.
  • Prove an angle in a semi-circle is a right angle.

Mathway Calculator Widget

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

Free Printable vectors Worksheets for 9th Grade

Math vectors made accessible! Discover a vast collection of free printable math vector worksheets, tailored for Grade 9 students and curated by Quizizz. Enhance your teaching experience today!

quizizz-hero

Explore vectors Worksheets by Grades

Explore other subject worksheets for grade 9.

  • social studies

Explore printable vectors worksheets for 9th Grade

Vectors worksheets for Grade 9 are an essential resource for teachers looking to enhance their students' understanding of math and algebra concepts. These worksheets provide a variety of exercises and problems that challenge students to apply their knowledge of vectors, helping them develop a strong foundation in this crucial area of mathematics. With a focus on Grade 9 level content, these worksheets are designed to align with the curriculum and learning objectives for this age group. Teachers can easily incorporate these resources into their lesson plans, using them as in-class activities, homework assignments, or assessment tools. By utilizing vectors worksheets for Grade 9, educators can ensure their students are well-prepared for more advanced math courses in the future.

In addition to vectors worksheets for Grade 9, teachers can also take advantage of Quizizz, an online platform that offers a wide range of interactive quizzes and activities to engage students in the learning process. Quizizz allows educators to create custom quizzes, drawing from a vast library of questions and resources related to math, algebra, and other subjects. This platform is an excellent complement to traditional worksheets, as it provides students with immediate feedback on their performance and helps teachers identify areas where additional support may be needed. Furthermore, Quizizz offers various game-like elements, making the learning experience more enjoyable and motivating for students. By incorporating both vectors worksheets for Grade 9 and Quizizz into their teaching strategies, educators can create a comprehensive and dynamic learning environment that supports student success in math and algebra.

  • International
  • Schools directory
  • Resources Jobs Schools directory News Search

Vectors in geometry

Vectors in geometry

Subject: Mathematics

Age range: 14-16

Resource type: Worksheet/Activity

SamSamson

Last updated

26 January 2014

  • Share through email
  • Share through twitter
  • Share through linkedin
  • Share through facebook
  • Share through pinterest

docx, 75.97 KB

Tes classic free licence

Your rating is required to reflect your happiness.

It's good to leave some feedback.

Something went wrong, please try again later.

Excellent, thank you. (Answers would have been even better!)

Empty reply does not make any sense for the end user

shelleygriffiths

Fantastic collection of questions that are slightly more stretching than the basic questions in vectors. Good for cementing understanding of the use of ratios in vectors. Thank you.

timfarmhouse

Fantastic Resource - exactly the practice my class needed.

Report this resource to let us know if it violates our terms and conditions. Our customer service team will review your report and will be in touch.

Not quite what you were looking for? Search by keyword to find the right resource:

IMAGES

  1. Vectors Worksheet With Answers

    geometry vectors worksheet

  2. Vectors Worksheet

    geometry vectors worksheet

  3. Vectors 2

    geometry vectors worksheet

  4. Vector Geometry Worksheet

    geometry vectors worksheet

  5. Resolving Vectors Worksheet With Answers

    geometry vectors worksheet

  6. Vector Math Practice Worksheet Answers

    geometry vectors worksheet

VIDEO

  1. Vectors worksheet Q. 20

  2. Geometry SecArc Worksheet

  3. 12X1 T03 04 geometric proofs 2024

  4. Worksheet Vectors Essay part SM IAU

  5. 3 dimensional geometry and vectors

  6. Vectors worksheet MCQ

COMMENTS

  1. Vector Geometry: Worksheets with Answers

    Mathster keyboard_arrow_up. Mathster is a fantastic resource for creating online and paper-based assessments and homeworks. They have kindly allowed me to create 3 editable versions of each worksheet, complete with answers. Worksheet Name. 1. 2. 3. Vectors on a grid. 1.

  2. Vector Worksheets

    Vector Worksheets Diverse and comprehensive, our printable vector worksheets successfully cater to high school students. A vector quantity has magnitude and direction. The right way to represent a vector is to draw it as a pointed arrow. The length of the arrow is the magnitude, and the arrowhead denotes the direction of the vector.

  3. Vectors Worksheets

    The easiest way to represent a vector is to draw an arrow. The length of the arrow represents the magnitude of the vector and the head represents the direction. Printable vectors worksheets. Find your perfect worksheet: visual, differentiated and fun. Includes a range of useful free math worksheets for kids.

  4. PDF VECTORS WORKSHEETS pg 1 of 13 VECTORS

    VECTORS Objectives Students will be able to: Define Sine, Cosine and Tangent in terms of the Use the above trig functions to finds angles and Define a vector in a sentence. Describe a vector's two main features. Define a scalar in a sentence. Give examples of vectors and scalars. Be able to identify if two vectors are equal

  5. Vectors

    Unit 1 Introduction to algebra Unit 2 Solving basic equations & inequalities (one variable, linear) Unit 3 Linear equations, functions, & graphs Unit 4 Sequences Unit 5 System of equations Unit 6 Two-variable inequalities Unit 7 Functions Unit 8 Absolute value equations, functions, & inequalities Unit 9 Quadratic equations & functions

  6. Vectors Practice Questions

    Click here for Answers. Practice Questions. Previous: Volume of a L-Shape Prism Practice Questions. Next: Use of a Calculator Practice Questions. The Corbettmaths Practice Questions on Vectors.

  7. PDF Two-Dimensional Vector Basics

    -2- Worksheet by Kuta Software LLC Find the magnitude and direction angle for each vector. 7) i j 8) r , Find the component form, magnitude, and direction angle for the given vector 9) CD where C = ( , ) D = ( , ) Sketch a graph of each vector then find the magnitude and direction angle.

  8. Vector Worksheet (pdf) with key. Focuses on resultant vectors. 25 problems

    Vectors Pictures of Vectors Law of Sines and Cosines (pre requisites for many of these problems) Resultant Vector worksheet (pdf) with answer key to all 25 problems on vectors and resultant vectors. Download this sheet for free

  9. Vectors

    Here we will learn about vectors, including what vectors are and how to use vectors to solve geometry problems. We will also learn how to add, subtract and multiply them. There are also vector worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you're still stuck. What are vectors?

  10. PDF Part I

    1. Find the length jvj and the angle for the vector 2. Write the vector below in therms of its compo-below. nents as v =< a; b >. 3. Below is a regular hexagon of each side equal to 2 units. First, use geometry to gure out the angle shown in the picture. Use the angle and the length 2 of each vector to write down the vectors u, v and w.

  11. PDF Exam Style Questions

    3. OABC is a trapezium. Point D is the midpoint of BC. Point E is the midpoint of AC. (a) Write these vectors in terms of a and b. 4. DFG is a straight line. Write down the vector. DF : FG = 2:3.

  12. Vectors Questions

    Revise Topic Specification Vectors Revision Vectors A vector is something with both magnitude and direction. On diagrams they are denoted by an arrow, where the length tells us the magnitude and the arrow tells us direction. You will need to add and subtract vectors You will also need to multiply vectors and understand scalar multiples of vectors

  13. PDF 4. Vector Geometry

    4.1. Vectors and Lines 211 Example 4.1.1 If v= 2 −1 3 then kvk= √ 4+1+9= √ 14. Similarly if v= 3 −4 in 2-space then kvk= √ 9+16=5. When we view two nonzero vectors as arrows emanating from the origin, it is clear geometrically what we mean by saying that they have the same or opposite direction. This leads to a fundamental new ...

  14. 50+ vectors worksheets on Quizizz

    Discover a vast collection of free printable math vectors worksheets, expertly crafted to help students master the fundamentals of vector operations and applications. Ideal for math teachers and learners alike. vectors Vectors 17 Q 9th - 12th Vectors 17 Q 8th - 9th Vectors 10 Q 11th - 12th Vectors 19 Q 11th Scalars/Vectors 11 Q 9th - 12th

  15. KS3 and KS4 Vectors Worksheets

    These worksheets are created in easy-to-download PDF format, include answers, and are designed to help your students better understand this complex concept of geometry. These vector worksheets are excellent resources that will make learning fun and exciting, helping your students improve at solving various critical functions related to vectors.

  16. Vector Problems

    Parallel vectors of the same magnitude are the same. These triangles are all identical therefore we can label the corresponding vectors aa and bb. 2 Decide the route. We need to find a route where we know the vectors. 3 Write the vector. → QT = − b − b + a − bQT = −b −b+ a−b. 4 Simplify your answer.

  17. Vector Worksheets

    Vector Worksheets Home > Topics > Vector Worksheets What are Vectors? Vectors is a term in science that denotes anything that has a magnitude as well as a direction. The right way to represent a vector is to draw it as a pointed arrow. The length of the arrow is the magnitude and the arrowhead points in the direction.

  18. 11.3E: Exercises for The Dot Product

    Answer: 22) a⇀ = i^ − j^, b⇀ = 7i^ + 2j^ −k^. 23) Find all two-dimensional vectors a⇀ orthogonal to vector b⇀ = 3, 4 . Express the answer in component form. Answer: 24) Find all two-dimensional vectors a⇀ orthogonal to vector b⇀ = 5, −6 . Express the answer by using standard unit vectors.

  19. PDF Print Layout

    Express the following vectors in terms of t and z. a) A C ... Solutions for the assessment Vector Geometry 1) a) P Q

  20. Vector Geometry (solutions, examples, videos)

    This video introduces Geometric Vectors, along with the magnitude, opposite vectors, congruent vectors, and resultants. A vector is a quantity that has both magnitude and direction. (It looks like a directed line segment). The length of a line segment is the magnitude. The direction indicates the direction of the vector.

  21. Vectors Geometry Worksheets & Teaching Resources

    Browse vectors geometry resources on Teachers Pay Teachers, a marketplace trusted by millions of teachers for original educational resources.

  22. Free Printable vectors Worksheets for 9th Grade

    By utilizing vectors worksheets for Grade 9, educators can ensure their students are well-prepared for more advanced math courses in the future. In addition to vectors worksheets for Grade 9, teachers can also take advantage of Quizizz, an online platform that offers a wide range of interactive quizzes and activities to engage students in the ...

  23. Vectors in geometry

    Worksheet of basic vectors, incuding adding vectors, finding vectors with midpoints or ratios of a line. Used at beginning of topic to ensure students understand direction and working with vectors. Please make comments. Report this resource to let us know if it violates our terms and conditions.