FOCUSED REVIEW
Focused Review — Vector Addition and Subtraction — Algebra-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: Algebra-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
Track component signs.
A⃗ = (2, −1) and B⃗ = (−3, 4). Find A⃗ + B⃗.
Reveal Answers
(−1, 3).
Why it works: Add x with x and y with y, including the signs.
ACTIVITY 2
Common Mistakes
Distribute the subtraction.
For A⃗ − B⃗, what happens to both components of B⃗?
Reveal Answers
Both are subtracted: (Aₓ − Bₓ, Aᵧ − Bᵧ).
Why it works: The minus sign acts on the entire vector.
ACTIVITY 3
Quick Application
Check the quadrant first.
If Rₓ < 0 and Rᵧ > 0, which quadrant contains R⃗?
Reveal Answers
Quadrant II.
Why it works: Component signs determine the quadrant before any inverse tangent is used.
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 Addition and Subtraction
Resolve each vector into x and y components, then combine corresponding components. This avoids scale-drawing error and works for any directions.
Rₓ = Aₓ ± Bₓ • Rᵧ = Aᵧ ± Bᵧ
Example: A⃗ = (5, 1), B⃗ = (2, −3): A⃗ − B⃗ = (3, 4).
Sensei note: Write parentheses around negative components before subtracting.
KEY CONCEPT 2
Magnitude and Direction from the Resultant
After finding Rₓ and Rᵧ, use the Pythagorean relationship for magnitude. Use the component signs to choose the correct quadrant for the direction angle.
|R⃗| = √(Rₓ² + Rᵧ²) • tan θ = Rᵧ / Rₓ
Example: For R⃗ = (3, 4), |R⃗| = 5 and θ = 53.1° above +x.
Sensei note: An inverse tangent alone can return an angle in the wrong quadrant.
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
Add components, then calculate magnitude and direction.
A⃗ = (3, 4) and B⃗ = (−1, 2). Find A⃗ + B⃗, its magnitude, and its direction from +x.
Reveal Answers
R⃗ = (2, 6); |R⃗| = √40 ≈ 6.32; θ ≈ 71.6° above +x.
Why it works: The component sum comes first; magnitude and direction are calculated from the resultant components.
PRACTICE 2
Independent Check
Subtract every component carefully.
A⃗ = (5, 1) and B⃗ = (2, −3). Find A⃗ − B⃗ and its magnitude.
Reveal Answers
(3, 4), with magnitude 5.
Why it works: The y-component is 1 − (−3) = 4, not −2.
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
Signed Components
Find the vector sum.
A⃗ = (−3, 2), B⃗ = (5, −6).
Reveal Answers
(2, −4).
Why it works: Add corresponding signed components.
QUICK CHECK 2
Vector Difference
Find the vector difference.
A⃗ = (4, 1), B⃗ = (−2, 3). Find A⃗ − B⃗.
Reveal Answers
(6, −2).
Why it works: Subtracting a negative x-component increases the x result: 4 − (−2) = 6.
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
Components First
Combine x-components and y-components separately before calculating any magnitude or angle.
KEY TAKEAWAY 2
Quadrant Matters
The signs of Rₓ and Rᵧ determine the resultant quadrant and prevent direction-angle errors.
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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