Find the Cartesian equation of a plane passing through a point and parallel to a given plane

by RoRi

May 30, 2016

Let $M$ be a plane which passes through the point $(1,2,-3)$ and is parallel to the plane given by the equation $3x-y+2z = 4$ . Find the Cartesian equation of $M$ . Further, find the distance between the two planes.

Let $M'$ denote the plane given by the equation $3x-y+2z = 4$ . Since $M$ and $M'$ are parallel, we know they share a common normal vector. Therefore, the Cartesian equation of $M$ is of the form

$3x - y + 2z = d.$

Since $(1,2,-3)$ is on $M$ , we have

$d = 3 - 2 - 6 = -5.$

Hence, the Cartesian equation of $M$ is

$3x - y + 2z = -5.$

The distance between $M$ and $M'$ is then

$\left| \frac{d_1 - d_2}{\lVert N \rVert} \right| = \frac{9}{\sqrt{14}}.$

Stumbling Robot

Find the Cartesian equation of a plane passing through a point and parallel to a given plane

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