How do I find the sine of the angle between two vectors?

1 Answer
Jul 28, 2018

The sine of the angle between vec(u) and vec(v) is:

(vec(u) xx vec(v))/(abs(u) abs(v))

Explanation:

I will assume you mean real valued two dimensional vectors..

Given vectors, vec(u) and vec(v), note that they can be represented in polar form as:

vec(u) = abs(u)((cos alpha) hat(i) + (sin alpha) hat(j))

vec(v) = abs(v)((cos beta) hat(i) + (sin beta) hat(j))

where alpha and beta are the angles that vec(u) and vec(v) make with the x axis.

Then:

vec(u) xx vec(v) = abs(u) (cos alpha) abs(v) (sin beta) - abs(u) (sin alpha) abs(v) (cos beta)

color(white)(vec(u) xx vec(v)) = abs(u) abs(v) (cos alpha sin beta - sin alpha cos beta)

color(white)(vec(u) xx vec(v)) = abs(u) abs(v) sin (beta - alpha)

So:

sin (beta - alpha) = (vec(u) xx vec(v))/(abs(u) abs(v))

which is the sine of the angle between the two vectors.

Three dimensions

For 3 dimensional vectors vec(u) and vec(v), the cross product is a vector quantity rather than a scalar one, but the absolute value of the sine of the angle between vec(u) and vec(v) is expressible in terms of the length of that vector quantity as:

(abs(vec(u) xx vec(v)))/(abs(u) abs(v))