Vector Angles in Radians

Question

Compute the angle in radians between each of the following vectors:

a) v = (-2, -2) and w = (2, -2)

b) v = (5, 0, 5) and w = (2, 2, 0)

c) v = (3, 3, 3, 3) and w = (0, 0, -1, -1)

Answer 1

a) cos∝ = v•w/(ιvι·ιwwι) = (-2·2 + (-2)·(-2))/(sqrt8·sqrt8) = 0; α = π/2;

b) cosφ = v•w/(ιvι· ιwι) = (5·2 + 0·2 + 5·0)/(sqrt(5² + 0² + 5²)·sqrt(2² + 2² + 0²)) = 10/20 = 0.5; φ = π/3

c)·cosθ = v•w/(ιvι· ιwι) = (3·0 + 3·0 + 3·(-1) + 3·(-1))/(sqrt(3² + 3² + 3² + 3²) · sqrt(0² + 0² +(-1)² + (-1)²)) = - 6/(6sqrt2 = - 1/sqrt2; θ = 3π/4