To determine the direction of the vector [tex]\(\vec{w} = \langle -1, 6 \rangle\)[/tex], we need to find the angle that the vector makes with the positive x-axis. We will use the arctangent function [tex]\( \arctan \)[/tex], specifically the [tex]\( \arctan2(y, x) \)[/tex], which takes into account the signs of both components to determine the correct quadrant of the angle. Here are the steps to find the direction:
1. Identify the components of the vector:
The vector [tex]\(\vec{w}\)[/tex] has components [tex]\(x = -1\)[/tex] and [tex]\(y = 6\)[/tex].
2. Calculate the angle in radians:
The angle [tex]\(\theta\)[/tex] in radians can be found using the function [tex]\(\arctan2(y, x)\)[/tex]:
[tex]\[
\theta = \arctan2(6, -1)
\][/tex]
This step corrects the angle based on the signs of both the x and y components, which places the angle in the correct quadrant.
3. Convert the angle from radians to degrees:
We usually want the angle in degrees for easier interpretation. The conversion from radians to degrees is done by multiplying the radian measure by [tex]\(\frac{180}{\pi}\)[/tex].
Using these calculations, for the vector [tex]\(\vec{w} = \langle -1, 6 \rangle\)[/tex]:
- The angle in radians is approximately [tex]\(1.7359450042095235\)[/tex] radians.
- The corresponding angle in degrees is approximately [tex]\(99.46232220802563^\circ\)[/tex].
Thus, the direction of the vector [tex]\(\vec{w} = \langle -1, 6 \rangle\)[/tex] is approximately [tex]\(99.46232220802563^\circ\)[/tex].
