Question

What angle θa, where 0∘≤θa<360∘, does a⃗ make with the +x-axis? θa = −66.8 ∘?

Answer 1

I think the thought of the question is to find a line that makes an angle of inclination of -66.8° to the x-axis. I'm guessing that would be a line or a tangent line to a curve, because angle is made up of two straight legs, with the one leg being the x-axis.

In analytical geometry, you can determine the angle of inclination to the x-axis using the slope of the line. The slope is the ratio of the change of y-coordinates to the change of x-coordinates: Δy/Δx. It is usually denoted as m. The relationship between m and θ is:

tan θ = m
 So, just substitute θ=−66.8°
tan −66.8° = m
m = -2.33

So, the slope is -2.33. The negative sign denotes that the slope is decreasing, so if you draw that, it would be a diagonal going down from left to right. Now, we use this value to the slope-intercept form of the line: y = mx +b. We know m but not b which is the y-intercept. We can't find the value of b because no additional data is given. So, let's just assume the line passes through the origin. Therefore, b = 0.

The equation of the line is y = -2.33x.