Question
Find the direction of bold u equals negative 3 bold i plus 8 bold j. (Round to the nearest tenth).
(1 point)
Responses
69.4°
69.4°
110.6°
110.6°
249.4°
249.4°
290.6°
(1 point)
Responses
69.4°
69.4°
110.6°
110.6°
249.4°
249.4°
290.6°
Answers
GPT-4o mini
To find the direction of the vector \(\mathbf{u} = -3 \mathbf{i} + 8 \mathbf{j}\), we can use the formula for the angle \(\theta\) in standard position, which is given by:
\[
\theta = \tan^{-1}\left(\frac{y}{x}\right)
\]
where \(y\) is the component in the \(j\) direction and \(x\) is the component in the \(i\) direction. In this case, \(x = -3\) and \(y = 8\).
1. Calculate \(\tan^{-1}\left(\frac{8}{-3}\right)\):
\[
\theta = \tan^{-1}\left(-\frac{8}{3}\right)
\]
2. Using a calculator, we find:
\[
\theta \approx -69.4^\circ
\]
This angle of \(-69.4^\circ\) is measured clockwise from the positive x-axis. To convert this angle to a positive angle in standard position (counterclockwise from the positive x-axis), we add \(360^\circ\):
\[
\theta = 360^\circ - 69.4^\circ = 290.6^\circ
\]
So, the direction of the vector \(\mathbf{u}\) is approximately:
\[
\boxed{290.6^\circ}
\]
\[
\theta = \tan^{-1}\left(\frac{y}{x}\right)
\]
where \(y\) is the component in the \(j\) direction and \(x\) is the component in the \(i\) direction. In this case, \(x = -3\) and \(y = 8\).
1. Calculate \(\tan^{-1}\left(\frac{8}{-3}\right)\):
\[
\theta = \tan^{-1}\left(-\frac{8}{3}\right)
\]
2. Using a calculator, we find:
\[
\theta \approx -69.4^\circ
\]
This angle of \(-69.4^\circ\) is measured clockwise from the positive x-axis. To convert this angle to a positive angle in standard position (counterclockwise from the positive x-axis), we add \(360^\circ\):
\[
\theta = 360^\circ - 69.4^\circ = 290.6^\circ
\]
So, the direction of the vector \(\mathbf{u}\) is approximately:
\[
\boxed{290.6^\circ}
\]