Question

Vector v has a direction of left angle bracket negative 3 comma 21 right angle bracket. Find the direction angle for v.
(1 point)
Responses

98.1°
98.1°

261.9°
261.9°

81.9°
81.9°

278.1°

Answer 1

To find the direction angle for the vector \( \mathbf{v} = \langle -3, 21 \rangle \), we can use the formula for the direction angle \( \theta \):

\[
\theta = \tan^{-1} \left( \frac{y}{x} \right)
\]

For the given vector \( \mathbf{v} = \langle -3, 21 \rangle \):

- \( x = -3 \)
- \( y = 21 \)

Substituting these values into the formula gives:

\[
\theta = \tan^{-1} \left( \frac{21}{-3} \right) = \tan^{-1} \left( -7 \right)
\]

Calculating \( \tan^{-1}(-7) \) will yield a value in the fourth quadrant (since \( -3 \) is negative and \( 21 \) is positive). The output from \( \tan^{-1} \) is generally from \(-90^\circ\) to \(90^\circ\). Therefore, we will need to adjust our angle to a positive angle in standard position.

Calculating that:

\[
\theta \approx -81.87^\circ
\]

To convert this to a positive angle, we add \( 360^\circ \):

\[
\theta = -81.87 + 360\approx 278.1^\circ
\]

Thus, the direction angle for the vector \( \langle -3, 21 \rangle \) is:

\[
\boxed{278.1^\circ}
\]