Vector Magnitude and Direction Math Example 5

Follow the full solution, then compare it with the other examples linked below.

Example 5

hard
Find the direction angle θ\theta of v=⟨−1,3⟩\mathbf{v} = \langle -1, \sqrt{3} \rangle measured from the positive xx-axis.

Solution

  1. 1
    tan⁡θ=3−1=−3\tan\theta = \frac{\sqrt{3}}{-1} = -\sqrt{3}. Since the vector is in Quadrant II, θ=120°\theta = 120°.
  2. 2
    Check: ⟨−1,3⟩\langle -1, \sqrt{3} \rangle points up-left, which is indeed Quadrant II ✓

Answer

120°120°
The direction angle is found using θ=arctan⁡(v2/v1)\theta = \arctan(v_2/v_1), adjusted for the correct quadrant. Always check which quadrant the vector lies in to avoid the wrong angle.

About Vector Magnitude and Direction

The magnitude ∥v∥\|\mathbf{v}\| is a vector's length; the direction is the angle it makes with a reference axis.

Learn more about Vector Magnitude and Direction →

More Vector Magnitude and Direction Examples

Example 1 easy

Find the magnitude of [formula].

Example 2 medium

Find the unit vector in the direction of [formula].

Example 3 medium

Find the magnitude and direction angle of the vector v = <3, 4>.

Example 4 easy

Find [formula].

← All Vector Magnitude and Direction Examples Vector Magnitude and Direction Concept