Three Dimensional Geometry

Find the distance of the point (3, 4, 5) from the plane - Wired Faculty

Question

Find the distance of the point (3, 4, 5) from the plane $straight r with rightwards arrow on top. space left parenthesis 2 space straight i with hat on top space minus space 5 space straight j with hat on top plus space 3 space straight k with hat on top right parenthesis space equals 13$

Solution

The equation of plane is

straight r with rightwards arrow on top. space left parenthesis 2 space straight i with hat on top space minus space 5 space straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis space equals space 13

or space left parenthesis 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 right parenthesis. space space left parenthesis 2 space straight i with hat on top space minus space 5 space straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis space equals space 13 or space space space 2 straight x minus 5 straight y plus 3 straight z space equals space 13 or space space space 2 straight x minus 5 straight y space plus space 3 straight z minus 13 space equals space 0

Let p be the length of perpendicular from point (3, 4, 5) to the plane.

therefore space space space straight p space space equals space fraction numerator open vertical bar 2 space left parenthesis 3 right parenthesis space minus space 5 space left parenthesis 4 right parenthesis space plus space 3 space left parenthesis 5 right parenthesis space space minus space 13 close vertical bar over denominator square root of 4 plus 25 plus 9 end root end fraction space equals space fraction numerator open vertical bar 6 minus 20 plus 15 minus 13 close vertical bar over denominator square root of 38 end fraction space equals space fraction numerator 12 over denominator square root of 38 end fraction