Question

Find the distance between the planes $straight r with rightwards arrow on top. space open parentheses straight i with hat on top space plus space 2 space straight j with hat on top space plus space space 3 space straight k with hat on top close parentheses space plus space 7 space equals 0$ and $straight r with rightwards arrow on top. space open parentheses 2 straight i with hat on top space plus space 4 space straight j with hat on top space plus space 6 space straight k with hat on top close parentheses space plus space 7 space equals space 0.$

Solution

The equations of the planes are

straight r with rightwards arrow on top. space open parentheses straight i with hat on top space plus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top close parentheses space plus space 7 space equals 0 space space and space straight r with rightwards arrow on top. space left parenthesis 2 space straight i with hat on top space plus space 4 space straight j with hat on top space plus space 6 space straight k with hat on top right parenthesis space plus space 7 space equals 0

open parentheses straight x stack straight i space with hat on top space plus space straight y space straight j with hat on top space plus space straight z space straight k with hat on top close parentheses. space space open parentheses straight i with hat on top space plus space 2 space straight j with hat on top space plus space space 3 space straight k with hat on top close parentheses space plus space 7 space equals space 0

and

open parentheses straight x space straight i with hat on top space plus space straight y space straight j with hat on top space plus space straight z space straight k with hat on top close parentheses space. space open parentheses 2 space straight i with hat on top space plus space 4 space straight j with hat on top space plus space 6 space straight k with hat on top close parentheses space plus space 7 space equals space 0

or space space straight x plus 2 straight y plus 3 straight z plus 7 space equals space 0

...(1)
and

2 straight x plus 4 straight y plus 6 straight z plus 7 space equals space 0

...(2)
These are parallel planes.
Let (x_1,y₁ , z₁) be any point on plane (1).
∴ x₁ + 2y₁ + 3z₁ + 7 - 0
or x₂ + 2y₁ + 3z₁ = –7 ...(3)
Length of perpendicular from (x₁ ,y₁ , z₁) to the plane (2) is

equals space open vertical bar fraction numerator 2 straight x subscript 1 plus space 4 straight y subscript 1 space plus space 6 space straight z subscript 1 plus 7 over denominator square root of 4 plus 16 plus 36 end root end fraction close vertical bar space equals space fraction numerator open vertical bar 2 space left parenthesis straight x subscript 1 space plus space 2 space straight y subscript 1 space plus space 3 space straight z subscript 1 right parenthesis plus 7 close vertical bar over denominator square root of 56 end fraction space equals space fraction numerator open vertical bar 2 space left parenthesis negative 7 right parenthesis space plus space 7 close vertical bar over denominator square root of 56 end fraction space equals space fraction numerator open vertical bar negative 14 plus 7 close vertical bar over denominator square root of 56 end fraction space equals space fraction numerator open vertical bar negative 7 close vertical bar over denominator square root of 56 end fraction space equals space fraction numerator 7 over denominator square root of 56 end fraction

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Three Dimensional Geometry

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