Precalculus / Unit 6: Applications of Trigonometry
6.5 The Dot Product and Angles Between Vectors
Notes
- $\langle a, b \rangle \cdot \langle c, d \rangle = ac + bd$. The result is a number, not a vector.
- Angle between vectors: $\displaystyle \cos\theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}||\mathbf{v}|}$.
- Two vectors are orthogonal (perpendicular) exactly when their dot product is 0.
- Example: $\langle 1, 0 \rangle \cdot \langle 1, 1 \rangle = 1$, and $\displaystyle \cos\theta = \frac{1}{1 \cdot \sqrt{2}}$, so $\theta = 45^\circ$.
Practice
What is $\langle 2, 3 \rangle \cdot \langle 4, -1 \rangle$?
$2(4) + 3(-1) = 5$.
For what value of $k$ are $\langle 3, k \rangle$ and $\langle 4, 6 \rangle$ orthogonal?
$12 + 6k = 0$ gives $k = -2$.
What is the angle between $\langle 1, 0 \rangle$ and $\langle 1, 1 \rangle$?
$\displaystyle \cos\theta = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$.
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