Vectors and the Geometry of Space

Question

Suppose v.w=8 and ||v × w ||=2, and the angle between v and w is θ. Find

(a) tanθ=

(b) θ=

Answer 1

The formula for the dot product of two vectors and the magnitude of cross product of two vectors is given by:

v.w = |v|*|w|*cos(theta) and ||v x w|| = |v|*|w|*sin(theta), where v and w are the two vectors.

So we have that a) tan(theta) = sin(theta)/cos(theta) = ||v x w|| / v.w = 2/8 = 1/4.

And b) theta = arctan(1/4).