​ Consider the vector equation of two planes .(2i + j + k) = 3, .(i – j – k) = 4 1. Find the vector equation of any plane through the intersection

Question

Consider the vector equation of two planes \(\bar r\).(2i + j + k) = 3, \(\bar r\).(i – j – k) = 4

1. Find the vector equation of any plane through the intersection of the above two planes. 
2. Find the vector equation of the plane through the intersection of the above planes and the point (1, 2, -1)

Answer 1

1. The cartesian equation are 2x + y + z – 3 = 0 and x – y – z – 4 = 0 Required equation of the plane is

(2x + y + z – 3) + λ(x – y – z – 4) = 0

(2+ λ)x + (1 – λ)y + (1 – λ)z + (-3 – 4λ) = 0.

2. The above plane passes through (1, 2, -1)

(2+ λ)1 + (1 – λ)2 + (1 – λ)(-1) + (-3 – 4λ) = 0

3 – 3 + 4λ = 0

λ = 0

Equation of the plane is 2x + y + z – 3 = 0

\(\bar r\) .(2i + j + k) = 3.