Question

How do I find the direction angle of vector `<-2, -5>`?

Answer 1

Step 1-
Decide from where you are going to measure your angle. Let's go with the convention: measuring a positive angle going counterclockwise from the positive x-axis.

Step 2-
Draw your vector!
`<-2,-5>` or `-2i-5j` is in the third quadrant. You go `2` units to the left on the `x` axis (in the negative `i` direction), and then from there down `5` units on the y axis (so below the origin).

Step 3-
Figure out the angle your vector makes with the x-axis, using some trig.

Step 4-
Figure out the overall angle starting from the positive x-axis from your sketch.

Now, you could actually have an infinite amount of solutions depending on where you are measuring your angle from (or you could just keep adding `360°` to get to the same place).

For example, another valid solution is to say that your direction angle, measured clockwise from the positive `x` axis is `360° - 248.2°= 141.6°`. Just make sure you specify what your frame of reference is.

For this case, I'm going to say the final answer is:

The direction angle for `<-2,-5>`, measured counterclockwise from the positive `x` axis, is `248.2°`