Question

What is the distance between two parallel planes?

Answer 1

Hint : First, we shall analyze the given information so that we are able to answer the given question. Here, we are asked to explain the distance between two parallel planes. The distance between the given two parallel planes is nothing but the shortest distance between the two plane surfaces.

Formula used:
The distance between two parallel planes, \[D\; = \;\dfrac{{|a{x_1}\; + \;b{y_1}\; + \;c{z_1}\; + \;d|}}{{\sqrt {{a_2}\;\; + \;{b_2}\; + \;{c_2}} }}\]

Complete step-by-step solution:
The distance between the given two parallel planes is nothing but the shortest distance between the two plane surfaces.
Let $a{x_1} + b{x_2} + c{x_3} - {d_1} = 0$ and $a{x_1} + b{x_2} + c{x_3} - {d_2} = 0$ .
Generally, two planes are said to be parallel if they don’t intersect each other and their ratios are equal.
The steps that are followed to determine the distance between two parallel planes are listed below.
a) We need to write the given equations for both the planes in standard form ${a_1}x + {b_1}y + {c_1}z + {d_1} = 0$and ${a_2}x + {b_2}y + {c_2}z + {d_2} = 0$ .
b) The next important step is to check whether the given equations are parallel or not. Generally, two planes are said to be parallel if they don’t intersect each other and their ratios are equal.
That is $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$
c) The next step is to pick the coefficients from the first plane equation. (i.e. $a,b,c,d$ )
d) Now, we shall identify the points from the second plane equation. (i.e. $x,y,z$ )
e) Now, we need to substitute the obtained values in the formula of distance.
Therefore, the required formula to calculate the distance between two parallel planes is as follows.
The distance between two parallel planes, \[D\; = \;\dfrac{{|a{x_1}\; + \;b{y_1}\; + \;c{z_1}\; + \;d|}}{{\sqrt {{a_2}\;\; + \;{b_2}\; + \;{c_2}} }}\]

Note: The distance between the given two parallel planes is nothing but the shortest distance between the two plane surfaces. After following the above steps, we need to substitute the values in the formula. Therefore, the required formula to calculate the distance between two parallel planes is as follows.
The distance between two parallel planes, \[D\; = \;\dfrac{{|a{x_1}\; + \;b{y_1}\; + \;c{z_1}\; + \;d|}}{{\sqrt {{a_2}\;\; + \;{b_2}\; + \;{c_2}} }}\]