1.

Find the angle between the lines `vecr=3hati-2hatj+6hatk+(lamda(2hati+hatj+2hatk))` and `vecr=(2hatj-5hatk)+(mu(6hati+3hatj+2hatk))`.

Answer» We have `vecr=3hati-2hatj|6hatk+lamda(2hati+hatj+2hatk)`
and `vecr=(2hatj-5hatk)+mu(6hati+3hatj+2hatk)`
where `veca_(1)=3hati-2hatj+6hatk,vecb_(1)=2hati+hatj+2hatk`
and `veca_(2)=2hatj-5hatk, vecb_(2)=6hati+3hatj+2hatk`
If `theta` is angle between the lines, then
`costheta=(|vec(b_(1)).vecb_(2)|)/(|vec(b_(1))|.|vec(b_(2))|)`
`=(|(2hati+hatj+2hatk).(6hati+3hatj+2hatk)|)/(|2hati+hatj+2hatk||6hati+3hatj+2hatk|)`
`=(|12+3+4|)/(sqrt(9)sqrt(49))=19/21`
`:. theta=cos^(-1)19/21`


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