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

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

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