Class 12

Math

3D Geometry

Three Dimensional Geometry

For the following planes, find the direction cosines of the normal to the plane and the distance of the plane from the origin.

$2x+3y−z=5$.

Given equation in vector form is $r⋅(2i^+3j^ −k^)=5$

$∣2i^+3j^ −k^∣=2_{2}+3_{2}+(−1)_{2} =14 $

$∴r⋅(14 2 i^+14 3 j^ −14 1 k^)=14 5 $

Direction cosines of $n^$(normal) are $14 2 ,14 3 ,14 −1 $ and $p=14 5 $.