$\vec{u}=(7,-4)$ $\newline$ Find the direction angle of $\vec{u}$ . $\newline$ Enter your answer as an angle in degrees between $0^{\circ}$ and $360^{\circ}$ rounded to the nearest hundredth. $\newline$ $\theta= \square \degree$ $\newline$

Full solution

\vec{u}=(7,-4)

\newline

Find the direction angle of

\vec{u}

\newline

Enter your answer as an angle in degrees between

0^{\circ}

and

360^{\circ}

rounded to the nearest hundredth.

\newline

\theta= \square \degree

\newline

Calculate Tangent: To find the direction angle of the vector $\vec{u} = (7, -4)$ , we need to calculate the angle $\theta$ that the vector makes with the positive x-axis. The direction angle can be found using the arctangent function $(\tan^{-1})$ , which gives us the angle whose tangent is the ratio of the y-coordinate to the x-coordinate of the vector.
Use Arctangent Function: Calculate the tangent of the angle $\theta$ using the y-coordinate and the x-coordinate of $\vec{u}$ : $\tan(\theta) = \frac{-4}{7}$ .
Perform Calculation: Use the arctangent function to find the angle $\theta$ : $\theta = \tan^{-1}\left(\frac{-4}{7}\right)$ . Make sure to use a calculator set to degree mode for this calculation.
Add $360$ Degrees: Perform the calculation: $\theta = \tan^{-1}(-4 / 7) \approx -29.74^\circ$ . This is the angle that $\vec{u}$ makes with the positive x-axis, measured counterclockwise. However, since the angle is negative, it is measured clockwise from the positive x-axis.
Round Direction Angle: To find the direction angle between $0^\circ$ and $360^\circ$ , we add $360^\circ$ to the negative angle: $\theta = -29.74^\circ + 360^\circ \approx 330.26^\circ$ .
Round Direction Angle: To find the direction angle between $0^\circ$ and $360^\circ$ , we add $360^\circ$ to the negative angle: $\theta = -29.74^\circ + 360^\circ \approx 330.26^\circ$ .Round the direction angle to the nearest hundredth: $\theta \approx 330.26^\circ$ .