Let u and v be two vectors in R^2 whose Euclidean norms satisfy |u| = 2|v|. What is the value of alpha such that w = u +...

Question

Let u and v be two vectors in R^2 whose Euclidean norms satisfy |u| = 2|v|. What is the value of alpha such that w = u + alpha*v bisects the angle between u and v?

A. 2
B. 1/2
C. 1
D. -1/2

Accepted Answer

A. 2 A vector w bisects the angle between u and v when the angle from u to w equals the angle from w to v. The standard result is that w = (u/|u|) + (v/|v|) — the sum of the unit vectors in the directions of u and v — bisects the angle between them. We must write this in the form u + alpha*v and find alpha.

Let u and v be two vectors in R^2 whose Euclidean norms satisfy |u| = 2|v|. What is the value of alpha such that w = u + alpha*v bisects the angle between u and v? A. 2 B. 1/2 C. 1 D. -1/2