FULL REVIEW
Full Review — Vector Addition and Subtraction — Algebra-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: 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 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 = (2, −1) and B⃗ = (−3, 4). Find A⃗ + B⃗.
Reveal Answers
(−1, 3).
Why it works: Corresponding components add independently.
ACTIVITY 2
Recall Activity 2
Answer before revealing the response.
A⃗ = (2, −1) and B⃗ = (−3, 4). Find A⃗ − B⃗.
Reveal Answers
(5, −5).
Why it works: Subtract each B component: 2 − (−3) = 5 and −1 − 4 = −5.
ACTIVITY 3
Recall Activity 3
Answer before revealing the response.
If Rₓ < 0 and Rᵧ > 0, which quadrant contains R⃗?
Reveal Answers
Quadrant II.
Why it works: Component signs determine the quadrant before the direction angle is interpreted.
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
Component-Wise Vector Operations
Once vectors are written in components, addition and subtraction are algebraic: operate on x-components together and y-components together. The signs retain the directional information.
Rₓ = Aₓ ± Bₓ • Rᵧ = Aᵧ ± Bᵧ
Example: A⃗ = (5, 1), B⃗ = (2, −3): A⃗ − B⃗ = (3, 4).
Sensei note: Use parentheses when a component is negative so subtraction signs are not lost.
KEY CONCEPT 2
Magnitude and Direction of a Resultant
After the resultant components are known, use the Pythagorean relationship for magnitude. Find a reference angle from the component ratio, then use signs to report the correct physical direction.
|R⃗| = √(Rₓ² + Rᵧ²) • tan θ = Rᵧ / Rₓ
Example: R⃗ = (2, 6) has |R⃗| = √40 ≈ 6.32 and θ ≈ 71.6° above +x.
Sensei note: Do not trust an inverse-tangent display until you have checked the quadrant.
KEY CONCEPT 3
Order and Grouping of Sums
Vector addition is commutative and associative. You may add several vectors in any order, but subtraction should first be rewritten as addition of the opposite vector.
A⃗ + B⃗ = B⃗ + A⃗ • (A⃗ + B⃗) + C⃗ = A⃗ + (B⃗ + C⃗)
Example: For three vectors, summing all x-components and all y-components gives the same result regardless of order.
Sensei note: Do not treat A⃗ − B⃗ as commutative; generally A⃗ − B⃗ ≠ B⃗ − A⃗.
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
Work from components to magnitude and direction.
A⃗ = (3, 4) and B⃗ = (−1, 2). Find A⃗ + B⃗, |R⃗|, and θ measured counterclockwise from +x.
Reveal Answers
R⃗ = (2, 6); |R⃗| = √40 ≈ 6.32; θ ≈ 71.6°.
Why it works: The resultant components are calculated first; the magnitude and direction follow from those components.
PRACTICE 2
Guided Problem
Distribute the subtraction to both components.
A⃗ = (5, 1), B⃗ = (2, −3). Find A⃗ − B⃗, |R⃗|, and its direction.
Reveal Answers
R⃗ = (3, 4); |R⃗| = 5; θ = 53.1° above +x.
Why it works: The y result is 1 − (−3) = 4, which places the resultant in Quadrant I.
PRACTICE 3
Independent Problem
Add all components before finding the final magnitude.
A⃗ = (4, −2), B⃗ = (−1, 5), C⃗ = (−2, −1). Find A⃗ + B⃗ + C⃗.
Reveal Answers
R⃗ = (1, 2); |R⃗| = √5 ≈ 2.24; θ ≈ 63.4° above +x.
Why it works: Summing components first avoids repeatedly converting between magnitude-angle and component forms.
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
Component Sum
Find the resultant.
A⃗ = (−3, 2), B⃗ = (5, −6). Find A⃗ + B⃗.
Reveal Answers
(2, −4).
Why it works: Add signed components independently.
QUICK CHECK 2
Component Difference
Find the resultant.
A⃗ = (4, 1), B⃗ = (−2, 3). Find A⃗ − B⃗.
Reveal Answers
(6, −2).
Why it works: The x-component is 4 − (−2) = 6; the y-component is 1 − 3 = −2.
QUICK CHECK 3
Magnitude and Direction
Find both quantities.
R⃗ = (3, −4). Find |R⃗| and describe its direction from +x.
Reveal Answers
|R⃗| = 5; 53.1° below +x, equivalently −53.1°.
Why it works: The negative y-component places the vector in Quadrant IV.
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
Operate Component by Component
Add or subtract corresponding components; signs carry direction information.
KEY TAKEAWAY 2
Reconstruct Magnitude and Direction
Use |R⃗| = √(Rₓ² + Rᵧ²) and determine the correct quadrant from component signs.
KEY TAKEAWAY 3
Addition Is Flexible; Subtraction Is Not
Vector sums may be reordered and regrouped, but subtraction should be converted to addition of the opposite vector.
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?
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