Find the angle between the following pair of lines: A lines with direction ratios 2,2,1 A line joning (3,1,4)to (7,2,12)

Options

A. `cos ^(-1)(2//3)`
B. `cos ^(-1)(3//2)`
C. `tan^(-1)(-2//3)`
D. None of the above

Answer

A

Explanation

Direction ratios of the line joining the point
`(3,1,4) and (7,2,12)` are
`=lt7-3,2-1,12-4gt`
`=lt4,1,8gt`
`=lea_(1),a_(2),a_(3)gt (let)`
Ant the directin ratio of given line is
`=lt2,2,1gt` lt brgt `=ltb_(1),b_(2),b_(3)gt` (let)
Let Q be the and between the lines,
then `cos theta=(a_(1)b_(1)+a_(2)b_(2)+a_(3)b_(3))/(sqrt(a_(1)^(2)+a_(2)^(1)+a_(3)^(2))sqrt(b_(1)^(2)+b_(2)^(2)+b_(3)^(2)))`
`impliescos theta=((4)(2)+(1)(2)+(8)(1))/(sqrt(16+1+64)sqrt(4+4+1))`
`impliescos theta=(18)/(sqrt81sqrt9)=(18)/(9xx3)=2/3`
`impliestheta=cos ^(-1)((2)/(3))`