Angle b/w 2 vectors u and v if the resultant of the 2 vectors = perpendicular to u and has magnitude half of v

Answer 1

The way to solve this problem follows from the law of sin and the las of cosines. The law of sin states:
a/sin A = b/sin B = c/sin C
by the law of cosines we know that:
c^2 = a^2 + b^2 - 2ab cos C
this may be a little hard to follow since i can't draw a picture in, but using the fact that cos(pi - t) = -cos(t), we can fill in your vectors now. BTW, the resultant (or c in this case) is determined by ||u + v||. Therefore:
||u + v||^2 = ||u||^2 + ||v||^2 + 2||u||||v||cos(t), where t is the angle between u and v, and from the law of sines:
||v|| / sin (A) = ||v + u|| / sin(pi - t), where A is the angle between u and the resultant.

since sin(pi - t) = sin(t), we can rearrange this formula to:
sin (A) = ( ||v|| / ||u + v|| ) sin (t)

now you know that the resultant is perpendicular to u, therefore A is 90 and sin(A) = 1. you also know that the resultant ||u + v|| has half the magnitude of ||v||, or ||u + v|| = (1/2) ||v||. from this it follows that:
1 = ( ||v|| / [(1/2)||v||] ) sin (t)

the ||v|| / ||v|| cancels out, and you are left with:
1 = 2sin(t)
sin(t) = 1/2
t = 30 degrees or 150

since the resultant is less than v, it must mean that angle between the two are obtuse, therefore
150 degrees

Answer 2

Let t be the angle between the vectors u and v.
Resolving perpendicular to u:
v sin(t) = v/2

Hence:
sin(t) = 1/2
t = 30deg or 150deg.

Answer 3

vcos(180 - θ) = u
vsinθ = v/2
sinθ = 1/2
θ = 150°