Question

The equation of a plane containing the point (1, - 1, 2) and perpendicular to the planes 2x + 3y - 2z = 5 and x + 2y - 3z = 8 is ______.

Options

• `"r"(4hat"i" - 5hat"j" + 3hat"k")` = 15

• `"r"(5hat"i" + 4hat"j" + 2hat"k")` = 5

• `"r"(5hat"i" + 4hat"j" - hat"k")` = 5

• `"r"(5hat"i" - 4hat"j" - hat"k")` = 7

Answer 1

The equation of a plane containing the point (1, - 1, 2) and perpendicular to the planes 2x + 3y - 2z = 5 and x + 2y - 3z = 8 is `underline("r"(5hat"i" - 4hat"j" - hat"k") = 7)`.

Explanation:

Given plane,

2K + 3y - 2z = 5 and x + 2y - 3z = 8

Normal of given plane are `2hat"i" + 3hat"j" - 2hat"k"` and `hat"i" + 2hat"j" - 3hat"k"` respectively,

Normal of plane perpendicular to given plane is,

`(2hat"i" + 3hat"j" - 2hat"k") xx (hat"i" + 2hat"j" - 3hat"k")`

`= |(hat"i", hat"j", hat"k"),(2,3,-2),(1,2,-3)| = - 5hat"i" + 4hat"j" + hat"k"`

Equation of plane passing through (1, - 1, 2) and whose normal vector `- 5hat"i" + 4hat"j" + hat"k"`

`"r" - (hat"i" - hat"j" + 2hat"k") * (- 5hat"i" + 4hat"j" + hat"k")` = 0

`"r" * (5hat"i" - 4hat"j" - hat"k") = - (- 5 - 4 + 2)`

`"r" * (5hat"i" - 4hat"j" - hat"k")` = 7