Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Dec 12, 2023

Find the direction of the vector with endpoint $(4,7)$ in standard position. The direction is $\arctan(7/4)^\circ$ (rounded to nearest hundredth).

STEP 1

Assumptions
1. The vector is in standard position, which means its initial point is at the origin $(0,0)$ .
2. The endpoint of the vector is $(4,7)$ .
3. The direction of the vector is the angle it makes with the positive x-axis, measured counterclockwise.
4. The direction angle is to be found in degrees.
5. The direction angle should be rounded to the nearest hundredth.

STEP 2

To find the direction of the vector, we need to calculate the angle it makes with the positive x-axis. The tangent of this angle is the ratio of the y-coordinate to the x-coordinate of the endpoint of the vector.
$\tan(\theta) = \frac{y}{x}$

STEP 3

Now, plug in the given coordinates for the endpoint of the vector into the equation.
$\tan(\theta) = \frac{7}{4}$

STEP 4

To find the angle $\theta$ , we need to take the arctangent (inverse tangent) of the ratio.
$\theta = \arctan\left(\frac{7}{4}\right)$

STEP 5

Use a calculator to find the arctangent of the ratio. Make sure the calculator is set to degree mode.
$\theta \approx \arctan\left(\frac{7}{4}\right)$

STEP 6

Calculate the angle to the nearest hundredth.
$\theta \approx \arctan\left(\frac{7}{4}\right) \approx 60.26^{\circ}$
The direction of the vector is approximately $60.26^{\circ}$ .

Was this helpful?

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Contact Influencer program Policy Terms