Topic 1.5: 2D Vector Addition

Name: _______________________________ Date: _______________

Before We Start: Yesterday's Recap

Yesterday we learned relative velocity in 1D. Without looking back: when do you ADD two velocities, and when do you SUBTRACT them?

Today's New Concepts

AP CED Alignment: Unit 1, Topic 1.5

Breaking a Vector into Components

• Any vector can be split into an x-component (horizontal) and a y-component (vertical)
• $x = A\cos\theta$, $y = A\sin\theta$ — θ measured from the positive x-axis (A = magnitude)
• Example: a vector of 8 m/s at 30° → $x = 8\cos(30°) \approx 6.9\text{ m/s}$, $y = 8\sin(30°) = 4\text{ m/s}$

Adding Vectors — Component Method

1. Break EVERY vector into x- and y-components
2. Add all the x-components together → total x
3. Add all the y-components together → total y
4. Recombine: $R = \sqrt{x^2+y^2}$, $\theta = \tan^{-1}\left(\dfrac{y}{x}\right)$

Worked Example — Walking 3 Blocks East, 4 Blocks North

• x-components: $3+0=3$. y-components: $0+4=4$
• Resultant magnitude: $\sqrt{3^2+4^2}=5\text{ blocks}$
• Direction: $\tan^{-1}\left(\dfrac{4}{3}\right) \approx 53°\text{ north of east}$
• This is the perpendicular special case — when vectors are already at 90°, you can skip straight to the Pythagorean theorem

Why This Matters Going Forward

This exact method — split into x and y, work each direction separately, recombine at the end — is how Days 15-16 handle projectile motion (horizontal motion and vertical motion are treated completely independently).

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Keep This Sheet!

These sheets build into your semester study guide. Keep them in order in a binder or folder — you'll want to flip back through them before quizzes, unit tests, and when AP exam review starts in the spring.