Question

Let `vec a ` be vector parallel to line of intersection of planes `P_1 and P_2` through origin. If `P_1`is parallel to the vectors `2 bar j + 3 bar k and 4 bar j - 3 bar k` and `P_2` is parallel to `bar j - bar k` and ` 3 bar I + 3 bar j `, then the angle between `vec a` and `2 bar i +bar j - 2 bar k` is :
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 1

Correct Answer - A
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`