IF planes `vec(r)*(phat(i)-hat(j)+2hat(k))+3=0andvec(r)*(2hat(i)-phat(j)-hat(k))-5=0` include angle `(pi)/(3)`, then the value of p is

Options

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

Answer

D

Explanation

We have,
`vec(r).(p hat(i)-hat(j)+2 hat(k))+3=0`
and `vec(r).(2hat(i)-p hat(j)- hat(k))-5=0`
Here, `vec(n)_(1)=p hat(i)- hat(j)+ 2hat(k)`
and `vec(n)_(2)=2hat(i)-p hat(j)- hat(k)`
Since, `cos theta= (vec(n)_(1)* vec(n)_(2))/(|vec(n)_(1)| |vec(n)_(2)|)`
`:.cos. (pi)/(3)= ((p hat(i)-hat(j)+2 hat(k))* ( 2 hat(i)-p hat(j)-hat(k)))/(sqrt(p^(2)+1+4)sqrt(4+p^(2)+1))` [ given, `theta=pi/3` ]
`1/2 = (2p+p-2)/(p^(2)+5)`
`rArr p^(2)+5=6p-4`
`rArr p^(2)-6p+9=0`
`rArr(p-3)^(2)=0`
`rArr p=3`