FULL REVIEW
Full Review — Vector Addition and Subtraction — Calculus-Based
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: Calculus-Based
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 = Aₓe₁ + Aᵧe₂ and B⃗ = Bₓe₁ + Bᵧe₂. Write A⃗ + B⃗.
Reveal Answers
(Aₓ + Bₓ)e₁ + (Aᵧ + Bᵧ)e₂.
Why it works: With a fixed basis, vector addition becomes addition of scalar coefficients.
ACTIVITY 2
Recall Activity 2
Answer before revealing the response.
r⃗₁ = (2, −1), r⃗₂ = (−1, 3). Find Δr⃗ = r⃗₂ − r⃗₁.
Reveal Answers
(−3, 4).
Why it works: Final minus initial gives the directed separation between the positions.
ACTIVITY 3
Recall Activity 3
Answer before revealing the response.
A⃗ = (1, 2), B⃗ = (−2, 1). Find 2A⃗ − B⃗.
Reveal Answers
(4, 3).
Why it works: Scalar multiplication and vector addition/subtraction distribute over components.
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
Vector Operations in a Fixed Basis
Choose a fixed basis and combine corresponding coefficients. The basis vectors themselves do not change during ordinary vector addition or subtraction, so the operation is linear and componentwise.
A⃗ ± B⃗ = (Aₓ ± Bₓ)e₁ + (Aᵧ ± Bᵧ)e₂
Example: A⃗ = (1, 2), B⃗ = (−2, 1): A⃗ + B⃗ = (−1, 3).
Sensei note: Do not mix components from different basis directions.
KEY CONCEPT 2
Displacement as a Position-Vector Difference
The displacement from one point to another is obtained by subtracting the initial position vector from the final position vector. A translation of the coordinate origin adds the same constant vector to both positions and therefore cancels.
Δr⃗ = r⃗₂ − r⃗₁
Example: r⃗₁ = (2, −1), r⃗₂ = (−1, 3): Δr⃗ = (−3, 4), |Δr⃗| = 5.
Sensei note: Order matters: reversing the subtraction reverses the displacement direction.
KEY CONCEPT 3
Linear Combinations Preserve Component Structure
Expressions such as αA⃗ + βB⃗ are evaluated by distributing the scalars and combining like basis components. This is the same algebra used later with vector-valued functions.
αA⃗ + βB⃗ = (αAₓ + βBₓ)e₁ + (αAᵧ + βBᵧ)e₂
Example: For A⃗ = (2, −1), B⃗ = (−3, 4), 3A⃗ − 2B⃗ = (12, −11).
Sensei note: Apply each scalar to every component of its vector before combining terms.
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
Evaluate the complete linear combination.
A⃗ = (2, −1), B⃗ = (−3, 4). Find 3A⃗ − 2B⃗ and its magnitude.
Reveal Answers
3A⃗ − 2B⃗ = (12, −11); magnitude = √265 ≈ 16.3.
Why it works: Scaling and subtraction are both componentwise; only after combining do you calculate the magnitude.
PRACTICE 2
Guided Problem
Use final position minus initial position.
r⃗₁ = 2e₁ − e₂ and r⃗₂ = −e₁ + 3e₂. Find Δr⃗ and |Δr⃗|.
Reveal Answers
Δr⃗ = −3e₁ + 4e₂; |Δr⃗| = 5.
Why it works: The basis coefficients subtract directly and the common coordinate origin cancels.
PRACTICE 3
Independent Problem
Subtract vector-valued functions before evaluating time.
r⃗₁(t) = (t², 2t) and r⃗₂(t) = (3t, t − 1). Find r⃗₂(t) − r⃗₁(t), then evaluate at t = 2.
Reveal Answers
r⃗₂ − r⃗₁ = (3t − t², −t − 1); at t = 2, the separation vector is (2, −3).
Why it works: Vector-valued functions add and subtract componentwise at every value of t.
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
Sum and Difference
Compute both vectors.
A⃗ = (1, −2), B⃗ = (4, 3). Find A⃗ + B⃗ and A⃗ − B⃗.
Reveal Answers
(5, 1) and (−3, −5), respectively.
Why it works: Corresponding components are combined independently.
QUICK CHECK 2
Position Difference
Find the displacement and magnitude.
r⃗₁ = (1, 2), r⃗₂ = (4, −2). Find r⃗₂ − r⃗₁.
Reveal Answers
(3, −4), with magnitude 5.
Why it works: Final minus initial gives the directed separation.
QUICK CHECK 3
Vector-Valued Function Sum
Evaluate at the stated time.
A⃗(t) = (t, t²), B⃗(t) = (2t, −t). Find A⃗(2) + B⃗(2).
Reveal Answers
(6, 2).
Why it works: At t = 2, A⃗ = (2, 4) and B⃗ = (4, −2); then add components.
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
Use a Fixed Basis Consistently
Add and subtract corresponding coefficients while the basis directions remain fixed.
KEY TAKEAWAY 2
Displacement Is a Difference
Δr⃗ = r⃗₂ − r⃗₁ is final minus initial and is unchanged by a common translation of the origin.
KEY TAKEAWAY 3
Linear Combinations Extend Naturally
Scalar multiples and vector-valued functions use the same componentwise addition and subtraction rules.
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
Great work!
You've completed this review. Choose the next resource that best matches how confident you feel.
I'm Still Unsure
Review the key ideas and examples again.
Review Again →
I Need More Practice
Continue with additional practice for this topic.
Go to Practice →
I'm Ready
Continue to the next recommended resource.
Continue →