Two angle between vectors questions please.?

Question

I keep getting both of these wrong and would appreciate direction and answers.

1. Find the angle between the following pair of vectors:
u= -i+4j, v=3i-4j

2.Write the following vector in the form ai+bj:Direction angle 240 degrees, magnitude 10

Many thanks!

Answer 1

Best to plot this on a piece of graph paper; "+i" is to the right' , "+j" is upwards, "-i" is to the left, "-j" is downwards, all with respect to the origin.

1. Draw the vectors on the graph paper. "u" is in the second quadrant (i is neg and j is pos). "v" is in the fourth quadrant, with (i is pos, j neg) . There are two answers to this question, since you could go from u to v or from u to v in the opposite direction. I always take the angle from zero to "u" and add it to the angle from 0 to "v". For "u", the tangent of the angle to the vertical axis is 1/4, then angle = 14%. So the angle from zero degrees to the vert axis = 90, and "u"s angle from zero degrees is 90 + 14 = 104 degrees. For "v", the tangent of the angle to the horizontal axis is 4/3 = 1.333.. and the angle = 53 degrees. Therefore the angle from "u" to "v" is 104 + 53 = 157 degrees. (the other answer is 360 degrees - 157 degrees = 203 degrees if you rotate in the other direction).

2. Again use graph paper. 240 degrees is 30 degrees less that 270 degrees (the vertical axis, down) draw a line 10 graph units long. It will be in the third quadrant where i and j will both be negative. At this point you can use a polar to rectangular converter, or do it by hand. By hand: the i side of the triangle formed with the down axis can be found by saying sin(angle) = i/10, and cos(angle) = j/10. The angle is 30 degrees (from vert downwards). so sin(30) = 0.5, and, cos(30) = 0.866; thus i= 0.5 and j = 0.866 (ignoring pos/neg). Since the vector is in the third quadrant, both i and j are negative.
Answer = -0.5i -0.866j

Hope this helps!

Answer 2

1)
cos(angle) = (u.v)/(|u||v|).
So u.v = -19, |u|=sqrt(17), |v| = 5. Substitute those all in and you get something like 157 degrees (or equivalently, 23 degrees).
2) This is just a right angled triangle. Draw a picture. You'll then see that you get a triangle with angle 60 degrees, with the negative i axis as one side, and negative j as the other. The i part is -10*cos(60), the j part is -10*sin(60).