2015-11-11 06:39:57
Determine the angles made by the vector v = (67)i + (-15)j with the positive x- and y-axes. write the unit vector n in the direction of v
2015-11-11 13:04:07

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: [latex]|v|= sqrt{ 67^{2} + 15^{2} }= sqrt{4489+225}= sqrt{4714}=68.8[/latex] 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