Let `vecA` be a vector parallel to the line of intersection of planes `P_(1) and P_(2)` . Plane `P_(1)` is parallel to vectors `2hatj + 3hatk and 4hatj - 3hatk nad P_(2) " is parallel to " hatj- hatk and 3hati + 3hatj`. Then the angle between vector `vecA and ` a given vector `2hati + hatj - 2hatk` is

Options

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

Answer

A

Explanation

Vectors parallel to the normals to `P_(1)` and `P_(2)` are
`vecn_(1)=(2hatj+3hatk)xx(4hatj-3hatk)` and `vecn_(2)=(hatj-hatk)xx(3hatj+3hatk)` respectively.
Clearly `vecn_(1)=-18hati` and `vecn_(2)=3hati-3hatj-3hatk`
`:.vecA=+-(vecn_(1)xxvecn_(2))=+-18hati(3hati-3hatj-3hatk)=+-54(hatj-hatk)`
Let `theta` be the angle between `vecA` and `vecn`. Then
`cos theta=(vecA.vecn)/(|vecA||vecn|)=+-(54(1+2))/(3xx54sqrt(2))=+-1/(sqrt(2))`
`impliestheta=(pi)/4` or `(3pi)/4`