Trigonometry / Unit 8: Oblique Triangles and Vectors
8.5 Vectors: Magnitude, Direction and Components
Notes
A vector has both magnitude (length) and direction. Written in component form, $\vec{v} = \langle x, y \rangle$.
- Magnitude: $\displaystyle |\vec{v}| = \sqrt{x^2 + y^2}$.
- Direction angle $\theta$ (from the positive $x$-axis): $\displaystyle \tan\theta = \frac{y}{x}$, then adjust for the quadrant.
- From magnitude and direction: $\vec{v} = \langle |\vec{v}|\cos\theta, |\vec{v}|\sin\theta \rangle$.
- Example: magnitude 6 at $60^\circ$ gives $\displaystyle \langle 3, 3\sqrt{3} \rangle$.
Practice
What is the magnitude of $\vec{v} = \langle -3, 4 \rangle$?
$\displaystyle \sqrt{9 + 16} = 5$.
A vector has magnitude 10 and direction angle $30^\circ$. What are its components?
$\displaystyle x = 10\cos 30^\circ = 5\sqrt{3}$ and $y = 10\sin 30^\circ = 5$.
What is the direction angle of $\vec{v} = \langle -1, 1 \rangle$, between $0^\circ$ and $360^\circ$?
The reference angle is $45^\circ$ and the vector is in Quadrant II, so $\theta = 180^\circ - 45^\circ = 135^\circ$.
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