Full solution

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

\newline

Find the direction angle of

\vec{u}

. Enter your answer as an angle in degrees between

0^{\circ}

and

360^{\circ}

rounded to the nearest hundredth.

\newline

\theta= \square^{\circ}

Calculate ratio of coordinates: To find the direction angle of the vector $\vec{u} = (7, -4)$ , we need to calculate the angle that this vector makes with the positive x-axis. The direction angle, often denoted as $\theta$ , can be found using the arctangent function (also known as the inverse tangent or $\text{atan}$ ), which gives us the angle whose tangent is the ratio of the y-coordinate to the x-coordinate of the vector.
Use arctangent function: First, we calculate the ratio of the y-coordinate to the x-coordinate of the vector $\vec{u}$ . This ratio is $-\frac{4}{7}$ . We will use this ratio to find the angle with the arctangent function.
Adjust angle for quadrant: Next, we use the arctangent function to find the angle $\theta$ . We have to be careful because the arctangent function will give us an angle in the range of $-90^\circ$ to $90^\circ$ , but we want the angle in the range of $0^\circ$ to $360^\circ$ . Since the x-coordinate is positive and the y-coordinate is negative, the vector is in the fourth quadrant. $\newline$ $\theta = \text{atan}(-\frac{4}{7})$
Calculate direction angle: Using a calculator, we find that $\theta \approx \text{atan}(-4/7) \approx -29.74^\circ$ . However, this is the angle measured counterclockwise from the positive x-axis to the vector, and since it's negative, it's actually measured clockwise. To find the direction angle in the range of $0^\circ$ to $360^\circ$ , we add $360^\circ$ to this angle. $\newline$ $\theta = 360^\circ - 29.74^\circ$
Add $360°$ for range adjustment: Performing the calculation gives us the direction angle of the vector in the correct range: $\newline$ $\theta = 360° - 29.74° \approx 330.26°$