​ Find the Cartesian and vector equations of the planes through the line of intersection of the planes r. (i - j) + 6 = 0 ​

Question

Find the Cartesian and vector equations of the planes through the line of intersection of the planes \(\bar{r}\). (\(\hat{i}\) - \(\hat{j}\)) + 6 = 0 and \(\bar{r}\). (3\(\hat{i}\) + 3\(\hat{j}\) - 4\(\hat{k}\)) = 0 which are at a unit distance from the origin.

Answer 1

Equation of plane through the line of intersection of two planes in vector form is

For the equation of plane Ax + By + Cz=D and point (x₁,y₁,z₁), a distance of a point from a plane can be calculated as

x.2 + y.1 + z.(-2)-3 = 0 

2x + y-2z-3 = 0 

For λ = -1

In Cartesian form are 2x + y - 2z - 3 = 0 & x + 2y-2z + 3 = 0