Question

What is the distance between the parallel planes \[3x + y - 4z = 2\] and \[3x + y - 4z = 24\] ?

Answer 1

Hint: In this problem, we need to find the distance between the two parallel planes equations are given, on comparing it with the general equations, \[{P_1} = ax + by + cz + {d_1} = 0\] and \[{P_2} = ax + by + cz + {d_2} = 0\] .We use the values into this formula for finding the distance between the two plane, \[d = \dfrac{{\left| {{d_1} - {d_2}} \right|}}{{\sqrt {{a^2} + {b^2} + {c^2}} }}\] .. We may derive a formula using this approach and use this formula directly to find the shortest distance between two parallel lines.

Complete step by step solution:
In the given problem,
The given two parallel planes are \[3x + y - 4z = 2\] and \[3x + y - 4z = 24\]
The parallel planes are in the form of
\[{P_1} = ax + by + cz + {d_1} = 0\] and \[{P_2} = ax + by + cz + {d_2} = 0\]
On comparing the equation of the plane \[{P_1} = ax + by + cz + {d_1} = 0\] with the given plane \[3x + y - 4z - 2 = 0\], we can get
 \[a = 3,b = 1,c = - 4\] and \[{d_1} = - 2\]
On comparing the equation of the plane \[{P_2} = ax + by + cz + {d_2} = 0\] with the given plane \[3x + y - 4z - 24 = 0\], we can get
 \[a = 3,b = 1,c = - 4\] and \[{d_2} = - 24\]
The formula for the distance between the parallel planes are \[d = \dfrac{{\left| {{d_1} - {d_2}} \right|}}{{\sqrt {{a^2} + {b^2} + {c^2}} }}\]
By substituting the value of \[a = 3,b = 1,c = - 4\], \[{d_1} = - 2\] and \[{d_2} = - 24\], we can get
 \[d = \dfrac{{\left| { - 2 - ( - 24)} \right|}}{{\sqrt {{3^2} + {1^2} + {{( - 4)}^2}} }}\]
 \[d = \dfrac{{\left| {22} \right|}}{{\sqrt {9 + 1 + 16} }}\]
By simplifying, we can get
 \[d = \dfrac{{22}}{{\sqrt {26} }}\]
Therefore, the distance between the two parallel planes is \[d = \dfrac{{22}}{{\sqrt {26} }}\] .
So, the correct answer is “ \[d = \dfrac{{22}}{{\sqrt {26} }}\] ”.

Note: We note that, if the two planes are parallel. Identify the coefficients \[a,{\text{ }}b,{\text{ }}c,\] and \[d\] from one plane equation. Find a point of the two planes. Substitute for \[a,{\text{ }}b,{\text{ }}c\] and \[d\] into the distance formula is \[d = \dfrac{{\left| {{d_1} - {d_2}} \right|}}{{\sqrt {{a^2} + {b^2} + {c^2}} }}\] .The shortest distance between two parallel lines is equal to determining how far apart lines are. This can be done by measuring the length of a line that is perpendicular to both of them. We may derive a formula using this approach and use this formula directly to find the shortest distance between two parallel lines.