Question

Find the angle between the pair of lines given by ` -> r=3 hat i+2 hat j-4 hat k+lambda( hat i+2 hat j+2 hat k)` and ` -> r=5 hat i-2 hat j+mu(3 hat i+2 hat j+6 hat k)` .

Answer 1

Let `theta` be the angle between the given lines.
The given lines are parallel to the vectors.
`bar(b_(1))=hat(i)+2hat(j)+2hat(k) and bar(b_(2))=3hat(i)+2hat(j)+6hat(k)` respectively.
`:.` The angle `theta` is given by
`cos theta =(bar(b_(1))*bar(b_(2)))/(|bar(b_(1))|*|bar(b_(2))|)`
`bar(b_(1))*bar(b_(2))=(hat(i)+2hat(j)+2hat(k))*(3hat(i)+2hat(j)+6hat(k))`
`1xx3+2xx2+2xx6`
`=3+4+12=19`
`bar(b_(1))=sqrt(1^(2)+2^(2)+2^(2))`
`=sqrt(9)=3`
`bar(b_(2))=sqrt(3^(2)+2^(2)+6^(2))`
`=sqrt(9+4+36)=sqrt(49)=7`
`:. cos theta = (19)/(3xx7)=(19)/(21)`
`:.theta=cos^(-1)((19)/(21))`