Consider the equation
`E_(1):vecr xx (2hati-hatj+3hatk)=3hati+hatk` and `E_(2): vecr xx (hati+2hatj-3hatk)=2hati-hatj`, then

Options

A. a) `E_(1)` represents a line
B. b) `E_(1)` represents two parallel planes
C. c) `E_(2)` represents a line
D. d) `E_(2)` represents two parallel planes

Answer

B, C, D

Explanation

`E_(1): vecr xx (2hati-hatj+3hatk)=3hati+hatk`
`rArr 3hati(3y+z)-hatj(3x-2z)+hatk(-x-2y)=3hati+hatk`
`therefore 3y+z=3, 3x-2y=0, -x-2y=1`
From first two equations, `3x-2(3-3y)=0`
`rArr 3x+6y=6 rArr x+2y=2`
Now, `x+2y=-1, x+2y=2` are parallel planes.
`E_(2): vecr xx (hati+2hatj-3hatk)=2hati-hatj`
`rArr hati(-3y-2z)-hatj(-3x-z)+hatk(2x-y)=2hati-hatj`
`therefore hati(-3y-2z)-hatj(-3x-z)+hatk(2x-y)=2hati-hatj`
`therefore -3y-2z=2, 3x+z=-1, 2x-y=0`
i.e., `-6x-2z=2, 3x+z=-1`
`therefore` Straight line, `2x-y=0, 3x+z=-1`