FOCUSED REVIEW
Focused Review — Vector Addition and Subtraction — Calculus-Based
Reinforce the highest-leverage ideas and representative problem-solving tools for Vector Addition and Subtraction.
TIME
Approximately 15 minutes
BEST FOR
Targeted reinforcement
FINISH WITH
A readiness check
After this focused review, you'll be able to...
reinforce the key relationships, apply them to representative problems, and identify what still needs work.
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 15 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
Key Ideas
Use the fixed basis.
A⃗ = Aₓe₁ + Aᵧe₂ and B⃗ = Bₓe₁ + Bᵧe₂. Write A⃗ + B⃗.
Reveal Answers
(Aₓ + Bₓ)e₁ + (Aᵧ + Bᵧ)e₂.
Why it works: A fixed basis lets corresponding scalar components combine independently.
ACTIVITY 2
Common Mistakes
Treat displacement as a difference.
r⃗₁ = (2, −1), r⃗₂ = (−1, 3). Find Δr⃗ = r⃗₂ − r⃗₁.
Reveal Answers
(−3, 4).
Why it works: Subtracting positions removes any common origin offset.
ACTIVITY 3
Quick Application
Evaluate a linear combination.
A⃗ = (1, 2), B⃗ = (−2, 1). Find 2A⃗ − B⃗.
Reveal Answers
(4, 3).
Why it works: Scale first or distribute componentwise; linearity gives the same result.
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
Reinforce the two highest-leverage relationships, then use them in representative situations.
KEY CONCEPT 1
Component-Wise Linearity
In a fixed basis, vector addition and subtraction act independently on corresponding scalar components. This structure extends immediately to scalar multiples and linear combinations.
A⃗ ± B⃗ = (Aₓ ± Bₓ)e₁ + (Aᵧ ± Bᵧ)e₂
Example: A⃗ = (1, 2), B⃗ = (−2, 1): 2A⃗ − B⃗ = (4, 3).
Sensei note: Keep the basis vectors fixed; combine coefficients, not basis directions.
KEY CONCEPT 2
Position Differences Produce Displacements
A displacement is the final position vector minus the initial position vector. Any constant shift of the coordinate origin cancels in the difference, so the displacement represents a geometric separation.
Δr⃗ = r⃗₂ − r⃗₁
Example: r⃗₁ = (2, −1), r⃗₂ = (−1, 3): Δr⃗ = (−3, 4), with magnitude 5.
Sensei note: Do not reverse the subtraction order; r⃗₂ − r⃗₁ and r⃗₁ − r⃗₂ point in opposite directions.
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
Apply the reinforced ideas to two representative situations, then use the strategy card to check your setup.
PRACTICE 1
Guided Example
Evaluate the linear combination component by component.
A⃗ = (2, −1), B⃗ = (−3, 4). Find 3A⃗ − 2B⃗.
Reveal Answers
(12, −11).
Why it works: 3A⃗ = (6, −3) and 2B⃗ = (−6, 8); subtracting gives (12, −11).
PRACTICE 2
Independent Check
Form the final-minus-initial difference.
r⃗₁ = 2e₁ − e₂ and r⃗₂ = −e₁ + 3e₂. Find Δr⃗ and its magnitude.
Reveal Answers
Δr⃗ = −3e₁ + 4e₂; |Δr⃗| = 5.
Why it works: The position difference is componentwise and the common origin does not appear in the result.
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
Vector Sum and Difference
Compute both.
A⃗ = (1, −2), B⃗ = (4, 3). Find A⃗ + B⃗ and A⃗ − B⃗.
Reveal Answers
A⃗ + B⃗ = (5, 1); A⃗ − B⃗ = (−3, −5).
Why it works: Each operation is applied independently to corresponding components.
QUICK CHECK 2
Displacement from Positions
Find the displacement.
r⃗₁ = (1, 2), r⃗₂ = (4, −2). Find r⃗₂ − r⃗₁ and its magnitude.
Reveal Answers
(3, −4), with magnitude 5.
Why it works: Final minus initial gives the directed separation; the magnitude follows from its 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
Linearity Is Component-Wise
Vector sums, differences, and scalar multiples can be evaluated by combining coefficients in a fixed basis.
KEY TAKEAWAY 2
Position Differences Are Geometric
Δr⃗ = r⃗₂ − r⃗₁ gives a displacement that is independent of a common shift of origin.
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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