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

Normal to plane `P_(1)` is
`vecn_(1)= (2hatj+3hatk)xx)(4hatj-3hatk)=-18hati`
Normal to plane `P_(2)` is
Therefore, `vecn_(2)= (hatj-hatk)xx (3hati +3hatj)=3hati -3hatj-3hatk`
`vecA` is parallel to `+-(vecn_(1)xx vecn_(2))=+- (-54hatj+54hatk)`
Now , the angle between `vecA nad 2hati +hatj - 2hatk` is given by
`cos thet=+-((-54hatj+54hatk).(2hati+hatj-2hatk))/(54sqrt2 .3)`
`=+-1/sqrt2`
`theta= pi//4 or 3pi//4`