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