Question

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

Answer 1

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))`