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

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))#