The angle between a line with direction ratios `lt 2, 2, 1gt` and a line joining the points `(3,1,4) and (7,2,12)` is

Options

A. `cos^(-1)(2/3)`
B. `cos^(-1)(-2/3)`
C. `tan^(-1)(2/3)`
D. none of these

Answer

A

Explanation

A line with direction ratios proportional to 2,2,1 is parallel to the vector `veca=2hati+2hatj+hatk`.
Line joining `P(3,1,4)` to `Q(7,2,12)` is parallel to the vector
`vec(PQ)=4hati+hatj+8hatk`
Let `theta` be the required angle. Then
`cos theta=(veca.vec(PQ))/(|veca||vec(PQ)|)=(8+2+8)/(sqrt(4|4|4)sqrt(16|1|64))`
`implies cos theta=18/(3xx9)=2/3impliestheta=cos^(-1)(2/3)`