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

##### 1 Answer

The sine of the angle between

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

#### Explanation:

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

Given vectors,

#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

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

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