Precalculus / Unit 6: Applications of Trigonometry
6.4 Vectors in the Plane
Notes
A vector has magnitude and direction. In component form $\langle a, b \rangle$:
- Magnitude: $\displaystyle |\mathbf{v}| = \sqrt{a^2 + b^2}$.
- Add or subtract component by component; scalar multiplication scales each component.
- A unit vector has magnitude 1: divide a vector by its magnitude. For $\langle 3, 4 \rangle$ it is $\displaystyle \langle \frac{3}{5}, \frac{4}{5} \rangle$.
- A vector with magnitude $r$ and direction angle $\theta$ is $\langle r\cos\theta, r\sin\theta \rangle$.
Practice
What is the magnitude of $\langle -5, 12 \rangle$?
$\displaystyle \sqrt{25 + 144} = 13$.
If $\mathbf{u} = \langle 2, 1 \rangle$ and $\mathbf{w} = \langle -1, 5 \rangle$, what is $3\mathbf{u} - \mathbf{w}$?
$\langle 6, 3 \rangle - \langle -1, 5 \rangle = \langle 7, -2 \rangle$.
What is the component form of a vector with magnitude 10 and direction angle $30^\circ$?
$\langle 10\cos 30^\circ, 10\sin 30^\circ \rangle$.
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