If planes `vecr*(p hati-hatj+2hatk)+3=0 and vecr*(2hati-phatj-hatk)-5=0` include angle `(pi)/(3)`, then the value of p

Options

A. 1,-3
B. `-1,3`
C. `-3`
D. 3

Answer

D

Explanation

We have,
`vecr*(phati-hatj+2hatk)+3=0`
and `vecr*(2hati-phatj-hatk)-5=0`
here, `vecn_(1)=phati-hatj+2hatk`
and `vecn_(2)=2hati-phatj-hatk`
since, `costheta=(vecn_(1)*vecn_(2))/(|vecn_(1)||vecn_(2)|)`
`therefore"cos"(pi)/(3)=((phati-hatj+2hatk)*(2hati-phatj-hatk))/(sqrt(p^(2)+1+4)sqrt(4+p^(2)+1))[given, theta=(pi)/(3)]`
`(1)/(2)=(2p+p-2)/(p^(2)+5)`
`impliesp^(2)+5=6p-4`
`impliesp^(2)-6p+9=0`
`implies(p-3)^(2)=0`
`impliesp=3`