AP Precalculus: Two-Dimensional Vectors Formulas & Properties
1. Magnitude of a Vector
- For \( \vec{v} = \langle a, b \rangle \), \( |\vec{v}| = \sqrt{a^2 + b^2} \)
2. Component Form
- From endpoints \( A(x_1, y_1) \), \( B(x_2, y_2) \):
\( \vec{AB} = \langle x_2-x_1, y_2-y_1 \rangle \)
3. Direction Angle
- \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \) for \( \vec{v} = \langle a, b \rangle \)
- Adjust \( \theta \) for correct quadrant
4. Component Form from Magnitude and Direction
- \( |\vec{v}| = r, \theta = \) direction angle:
- Component form: \( \vec{v} = \langle r\cos\theta, r\sin\theta \rangle \)
5. Resultant Vector (Sum) Methods
- Triangle Method: Tail of second vector at tip of first; sum is vector from start to end point.
- Parallelogram Method: Draw vectors from common point; diagonal is resultant.
6. Vector Addition & Subtraction
- Add: \( \langle a, b \rangle + \langle c, d \rangle = \langle a + c, b + d \rangle \)
- Subtract: \( \langle a, b \rangle - \langle c, d \rangle = \langle a - c, b - d \rangle \)
7. Magnitude & Direction of Vector Sum
- Add components as above to get resultant \( \vec{R} \)
- \( |\vec{R}| = \sqrt{(a+c)^2 + (b+d)^2} \)
- \( \theta_R = \tan^{-1}\left(\frac{b+d}{a+c}\right) \)
8. Scalar Multiplication
- \( k \cdot \langle a, b \rangle = \langle ka, kb \rangle \)
- |k·v| = |k||v|; direction unchanged or reversed if \( k < 0 \)
9. Unit Vector
- Unit vector: magnitude 1, same direction as \( \vec{v} \)
- \( \mathbf{u} = \frac{\vec{v}}{|\vec{v}|} \)
10. Linear Combinations
- \( a\vec{v} + b\vec{w} \) = \( a\langle v_1, v_2 \rangle + b\langle w_1, w_2 \rangle = \langle av_1 + bw_1, av_2 + bw_2 \rangle \)