Distance Between Parallel Planes

Here you will learn how to find distance between parallel planes with examples.

Let’s begin –

Distance Between Parallel Planes

Let ax + by + cz + $d_1$ = 0 and ax + by + cz + $d_2$ = 0 be two parallel planes. In order to find the distance between them, we may follow the following algorithm.

Algorithm :

1). Take an aribitrary point P$(x_1, y_1, z_1)$ on one of the planes, say ax + by + cz + $d_1$ = 0.

2). Find the length of the perpendicular ‘d’ drawn form P $(x_1, y_1, z_1)$ on the other plane i.e ax + by + cz + $d_2$ = 0. Clearly,

d = |$ax_1 + by_1 + cz_1 + d_2\over \sqrt{a^2 + b^2 + c^2}$|

3). As P$(x_1, y_1, z_1)$ lies on the plane ax + by + cz + $d_1$ = 0.

$\therefore$  $ax_1 + by_1 + cz_1 + d_1$ = 0 $\implies$ $ax_1 + by_1 + cz_1$ = $-d_1$

4). Substitute $ax_1 + by_1 + cz_1$ = $-d_1$ in the expression for d obtained in step 2 to get d = $|d_2 – d_1|\over \sqrt{a^2 + b^2 + c^2}$, which gives the required distance.

Remark 1 : So the formula used to find the distance between the parallel planes ax + by + cz + $d_1$ = 0 and ax + by + cz + $d_2$ = 0 is

d = $|d_1 – d_2|\over \sqrt{a^2 + b^2 + c^2}$

Remark 2 : The distance between the parallel planes ax + by + cz + $d_1$ = 0 and $\lambda$(ax + by + cz) + $d_2$ = 0 is given by

d = $|d_1 – {d_2/\lambda}|\over \sqrt{a^2 + b^2 + c^2}$

Example : Find the distance between the parallal planes x + y – z + 4 = 0 and x + y – z + 5 = 0.

Solution : Here, $d_1$ = 4 and $d_2$ = 5

So, d = $|d_1 – d_2|\over \sqrt{a^2 + b^2 + c^2}$

= $|4 – 5|\over \sqrt{3}$ = $1\over \sqrt{3}$