The equation of the plane passing through the point ( 1,1, 1 ) and...

The equation of the plane passing through the point

( 1,1, 1 )

and perpendicular to the planes

2x + y - 2z = 5

and

3x - 6y - 2z = 7

Options

A $14x + 2y + 15z = 31$
B $14x + 2y - 15z = 1$
C $- 14x + 2y + 15z = 3$
D $14x - 2y + 15z = 27$

Correct answer

A. $14x + 2y + 15z = 31$

Step-by-step solution

The normal vector of required plane is parallel to vector

$= \left|$ \begin{cases}\overset{^}{i} & \overset{^}{j} & \overset{^}{k} \\ 2 & 1 & - 2 \\ 3 & - 6 & -2\end{cases} $\right| = - 14\overset{^}{i} - 2\overset{^}{j} - 15\overset{^}{k}$

$∴$ The equation of required plane passing through (1, 1, 1) will be $- 14\left(x - 1\right) - 2\left(y - 1\right) - 15\left(z - 1\right) = 0$

$⇒ 14x + 2y + 15z = 31$

$∴$ option $14x + 2y + 15z = 31$ is correct.

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