Let ` vec A` be a vector parallel to the line of intersection of planes `P_1a n dP_2dot` Plane `P_1` is parallel to vectors `2 hat j+3 hat ka n d4 hat j-3ka n dP_2` is parallel to ` hat j- hat ka n d3 hat i+3 hat jdot` Then the angle betweenvector ` vec A` and a given vector `2 hat i+ hat j-2 hat k` is `pi//2` b. `pi//4` c. `pi//6` d. `3pi//4`

Options

A. `(pi)/(2)`
B. `(pi)/(4)`
C. `(pi)/(6)`
D. `(3pi)/(4)`

Answer

B, D

Explanation

Let vector `vec(AO) ` be parallel to line of intersection of planes `P_(1) " and " P_(2) ` through origin .
Normal to plane `P_(1)` is
` vec(n)_(1) =[(2hat(j) + 3hat(k)) xx (4hat(j) - 3hat(k))] =- 18hat(j)`
So , `vec(OA) ` is parallel to `+- (vec(n)_(1) xx vec(n)_(2)) = 54 hat(j) - 54 hat(k)`
`:. ` Angle between `54 (hat(j) - hat(k)) " and " (2hat(i) + hat(j) - 2hat(k)) ` is
`" cos" 0 = +- ((54 + 108)/(3.54 "."sqrt(2))) =+- (1)/(sqrt(2))`
` :. 0 = (pi)/( 4) , ( 3pi)/(4)`
Hence (b) and (d) are correct answers