Let `P_(1)` denote the equation of a plane to which the vector `(hati+hatj)` is normal and which contains the line whose equation is `vecr=hati+hatj+veck+lamda(hati-hatj-hatk)andP_(2)` denote the equation of the plane containing the line L and a point with position vector j. Which of the following holds good? (a) The equation of `P_(1)" is"x+y=2.` (b) The equation of `P_(2)" is "vecr.(hati-2hatj+hatk)=2.` (c) The acute angle the `P_(1)andP_(2)" is "cot^(-1)(sqrt3)`. (d) The angle between the plnae `P_(2)` and the line L is `tan^(-1)sqrt3`.

Options

A. The equation of `P_(1)" is"x+y=2.`
B. The equation of `P_(2)" is "vecr.(hati-2hatj+hatk)=2.`
C. The acute angle the `P_(1)andP_(2)" is "cot^(-1)(sqrt3)`.
D. The angle between the plnae `P_(2)` and the line L is `tan^(-1)sqrt3`.

Answer

a, c

Explanation

Plane `P_(1)` contains the line
`vecr=hati+hatj+hatk+lamda(hati-hatj-hatk)`, hence contains the point `hati+hatj+hatk` and is normal to vector `(hati+hatj)`.
Hence, equation of plane is
`" "(vecr-(hati+hatj+hatk))*(hati+hatj)=0`
or `" "x+y=2`
Plane `P_(2)` contains the line
`" "vecr=hati+hatj=hatk+lamda(hati-hatj-hatk)` and point `hatj`
Hence, equation of plane is
`|{:(x-0,,y-1,,z-0),(1-0,,1-1,,1-0),(1,,-1,,-1):}|=0`
or `" "x+2y-z=2`
If `theta` is the acute angle between `P_(1) and P_(2)`, then
`" "costheta= (vec(n_1)*vec(n_2))/(|vec(n_1)||vec(n_2)|)=|((hati+hatj)*(hati+2hatj-hatk))/(sqrt2*sqrt6)|`
`" "= (3)/(sqrt2*sqrt6)= (sqrt(3))/(2)`
`" "theta=cos^(-1)""(sqrt(3))/(2)= (pi)/(6)`
As `L` is containec in `P_(2) rArr theta=0`