A non-zero vectors a is parallel to the line of intersection of the plane determined by the vectors `hat(i), hat(i)+hat(j) and ` the plane determined by the vectors `hat(i)-hat(j) and hat(i)+hat(k)`. Find the angle between a and `hat(i)-2hat(j)+2hat(k)`.

Options

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

Answer

D

Explanation

The line of intersection of the two planes is perpendicular to their normals. So, it parallel to the vector
`veca={(hatixx(hati+hatj)}xx{(hati-hatj)xx(hati=hatk)}`
`impliesveca=hatkxx(-hatj+hatk-hatk)=hati-hatj`
Let `theta` be the angle between `veca` and `vecb-hati-2hatj+2hatk`. Then
`cos theta=(veca.vecb)/(|veca||vecb|)=3/(sqrt(2)xx3)=1/(sqrt(2))impliestheta=(pi)/4`