Question

If θ is the angle between two vectors `hati - 2hatj + 3hatk and 3hati - 2hatj + hatk` find `sin theta`

Answer 1

Let `veca = hati - 2hatj + 3hatk` and `vecb = 3hati - 2hatj + hatk`. if `theta` is the angle between them. Then

`cos theta = (veca.vecb)/(|veca||vecb|)`

Now,

`veca.vecb = (veci - 2hatj + 3hatk).(3hati - 2hatj + hatk)`

= 3 + 4 + 3 = 10

`|veca| = sqrt(1+4+9) = sqrt(14)` and `|vecb| = sqrt(9 + 4 + 1) = ssqrt14`

`:. cos theta = (veca.vecb)/(|veca||vecb|)`

`=> cos theta = 10/(sqrt14sqrt14) = 10/14 = 5/7`

Now

`sintheta = sqrt(1-cos^2 theta)`

`=sqrt(1- (5/7)^2)`

`= sqrt24/49 = (2sqrt6)/7`