To determine the direction in which vector v points, we can calculate its angle with respect to the positive x-axis. The vector v is given by $\vec{v} = \langle -10, 10 \rangle$, where the x-component is -10 and the y-component is 10.

Step 1: Calculate the angle (θ)

The direction of the vector can be found using the formula for the angle θ between the vector and the positive x-axis:

$\theta = \tan^{-1}\left(\frac{y}{x}\right)$

For $\vec{v} = \langle -10, 10 \rangle$:

$\theta = \tan^{-1}\left(\frac{10}{-10}\right) = \tan^{-1}(-1)$

$\theta = -45^\circ$

Step 2: Adjust for the quadrant

Since the vector has a negative x-component and a positive y-component, it lies in the second quadrant. To find the actual angle from the positive x-axis, we add 180° to the calculated angle:

$\theta = 180^\circ - 45^\circ = 135^\circ$

Conclusion:

The vector $\vec{v} = \langle -10, 10 \rangle$ points at an angle of 135° counterclockwise from the positive x-axis.

Would you like further details or have any questions?

Here are 5 related questions:

1. What is the magnitude of vector $\vec{v}$?
2. How would the direction change if the x-component were positive?
3. Can you find the unit vector in the direction of $\vec{v}$?
4. How can you express the vector $\vec{v}$ in polar coordinates?
5. What would be the result if you added another vector to $\vec{v}$?

Tip: The quadrant a vector lies in is determined by the signs of its x- and y-components.