Consider the picture attached.
From right triangle trigonometry:
tan(α)=(opposite side)/(adjacent side)=15/67=0.2239
using a scientific calculator we find that arctan(0.2239)=12.62°
thus α=12.62°, is the angle that the vector makes with the positive x-axis.
The angle made with the + y-axis is 12.62°+90°=102.62°.
The length of the vector v can be determined using the Pythagorean theorem:
[tex]|v|= \sqrt{ 67^{2} + 15^{2} }= \sqrt{4489+225}= \sqrt{4714}=68.8[/tex]
Thus, to make v a unit vector, without changing its direction, we need to divide v by |v|=68.8.
This means that the x and y components will also be divided by 68.8, by proportionality.
So, the unit vector in the direction of v is:
(67/68.8)i + (-15/68.8)j=0.97 i + (- 0.22)j
Answer: 12.62°; 102.62°; 0.97 i + (- 0.22)j
