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

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