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.