FULL REVIEW
Full Review — Vector Addition and Subtraction — Foundational
Review the essential ideas, relationships, and problem-solving tools for Vector Addition and Subtraction.
TIME
45–60 minutes
BEST FOR
A complete topic review
FINISH WITH
A readiness check
After this full review, you'll be able to...
recall the essential ideas, apply them to representative problems, and determine what to study next.
Choose how you want to review
Course Alignment
This Physics Sensei Topic Review is an independent learning resource. Use it to reinforce key concepts, prepare for homework, or review before a quiz or exam.
Related Unit Review: If you need to review the complete unit material, review Vectors and Components here →
RESOURCE: Physics Sensei Topic Review | TOPIC ID: MEC-U02-T02 | TOPIC: Vector Addition and Subtraction | PARENT UNIT: MEC-U02 — Vectors and Components | COURSE LEVEL: Foundational
BEST USED ✓ After learning the topic ✓ Before starting homework ✓ Before a quiz or exam
Your Review Plan
Complete these six stages in order. Each stage builds on the previous one and prepares you for the final readiness check.
6 Stages • Approximately 45–60 minutes.
Warm-Up Check
Before you begin, take a moment to see what you already remember. Do not worry about getting everything right. This is only a starting point.
ACTIVITY 1
Recall Activity 1
Answer before revealing the response.
A⃗ points 4 units east and B⃗ points 3 units north. Sketch A⃗ + B⃗ and describe its direction.
Reveal Answers
The resultant points northeast from the starting tail to the final head.
Why it works: The head-to-tail construction shows the net change directly.
ACTIVITY 2
Recall Activity 2
Answer before revealing the response.
If C⃗ points south, which way does −C⃗ point?
Reveal Answers
North, with the same magnitude as C⃗.
Why it works: Multiplying a vector by −1 reverses its direction without changing its magnitude.
ACTIVITY 3
Recall Activity 3
Answer before revealing the response.
A⃗ and B⃗ have equal magnitude and point in opposite directions. What is A⃗ + B⃗?
Reveal Answers
The zero vector.
Why it works: Equal opposite vectors cancel because their directed changes are exact opposites.
Ready to strengthen your understanding?
You've refreshed what you already know. Next, you'll reinforce the essential concepts that will help you solve problems with confidence. Need to see the learning path again?
Core Concepts
Let's rebuild the key ideas one step at a time. Focus on understanding the relationships before worrying about solving problems.
KEY CONCEPT 1
Head-to-Tail Vector Addition
A vector can be translated without changing it. Place the tail of each new vector at the head of the previous one. The resultant connects the original starting point to the final endpoint.
R⃗ = A⃗ + B⃗
Example: A 4 m east displacement followed by 3 m north gives a resultant that closes the triangle from start to finish.
Sensei note: Do not rotate a vector when moving it into head-to-tail position.
KEY CONCEPT 2
Subtraction Is Addition of an Opposite
The negative of a vector has the same magnitude and the opposite direction. Replace subtraction by addition of that reversed vector, then use the ordinary addition rule.
A⃗ − B⃗ = A⃗ + (−B⃗)
Example: If B⃗ is 2 N west, then −B⃗ is 2 N east; A⃗ − B⃗ therefore adds an eastward contribution.
Sensei note: Reverse the entire vector. A minus sign does not mean “ignore direction.”
KEY CONCEPT 3
Resultants Can Be Checked Geometrically
For perpendicular vectors, the resultant forms the hypotenuse of a right triangle. A scale drawing can also estimate the magnitude and direction before calculation.
R² = A² + B² (perpendicular vectors)
Example: A 5 m east vector plus a 12 m north vector has resultant magnitude 13 m.
Sensei note: Use the right-triangle relationship only when the two vectors are perpendicular.
Ready to apply these ideas?
You've reinforced the essential concepts. Now it's time to put them into practice by working through guided examples and building your problem-solving confidence. Need a quick reminder?
Guided Practice
Now it's time to apply what you've reviewed.
Work through each activity in order. The examples become gradually more challenging, and each one prepares you for the final readiness check.
PRACTICE 1
Worked Example
Sketch first, then calculate.
A student walks 3 m east and 4 m north. Find the resultant displacement magnitude and describe its direction.
Reveal Answers
5 m toward the northeast.
Why it works: The vectors are perpendicular, so the resultant is the 5 m hypotenuse of a 3–4–5 triangle.
PRACTICE 2
Guided Problem
Reverse the vector being subtracted.
A⃗ is 8 N east and B⃗ is 3 N east. Find A⃗ − B⃗.
Reveal Answers
5 N east.
Why it works: Subtracting 3 N east is equivalent to adding 3 N west.
PRACTICE 3
Independent Problem
Combine the sequence into one net change.
A hiker goes 5 m east, 12 m north, then 5 m west. Find the final displacement from the starting point.
Reveal Answers
12 m north.
Why it works: The 5 m east and 5 m west vectors cancel, leaving only the northward displacement.
Ready to check your understanding?
You've practiced the essential skills with guidance. Now it's time to solve a few short problems on your own and confirm you're ready to move forward. Need a quick reminder?
Confidence Check
You've rebuilt the key ideas and practiced them with guidance. Now try these short questions on your own to check your understanding before moving on.
QUICK CHECK 1
Cancellation Check
Give magnitude and direction.
What is the sum of 6 N east and 6 N west?
Reveal Answers
0 N; the resultant is the zero vector.
Why it works: Equal opposite vectors cancel exactly.
QUICK CHECK 2
Difference Check
Use addition of the opposite.
A⃗ is 9 units north and B⃗ is 4 units north. What is A⃗ − B⃗?
Reveal Answers
5 units north.
Why it works: Subtracting B⃗ adds 4 units south to the 9-unit north vector.
QUICK CHECK 3
Perpendicular Resultant
Find the magnitude.
A⃗ is 8 units east and B⃗ is 15 units north. What is |A⃗ + B⃗|?
Reveal Answers
17 units.
Why it works: The vectors are perpendicular, so the magnitude follows the 8–15–17 right triangle.
How did it go?
You've checked your understanding. Take one final look at the essential ideas before deciding what to do next. Need a quick reminder?
Summary
Before moving on, take one final look at the most important ideas from this review.
KEY TAKEAWAY 1
Addition Represents Net Change
Use head-to-tail construction and draw the resultant from the first tail to the last head.
KEY TAKEAWAY 2
Subtraction Reverses a Vector
A⃗ − B⃗ means A⃗ + (−B⃗); the opposite vector has equal magnitude and opposite direction.
KEY TAKEAWAY 3
Check Geometry Before Calculating
Estimate direction first and use right-triangle geometry only when the vectors are perpendicular.
Ready for your next step?
You've reviewed the essential ideas one last time. Now choose the resource that best matches how confident you feel. Need a quick reminder?
Next Step
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