​ Find the vector equation of the plane passing through the intersection of the planes r.(2i- 7j + 4k) = 3 and r.(3i - 5j + 4k) + 11 = 0, ​

Question

Find the vector equation of the plane passing through the intersection of the planes \(\bar{r}\).(2\(\hat{i}\)- 7\(\hat{j}\) + 4\(\hat{k}\)) = 3 and \(\bar{r}\).(3\(\hat{i}\) - 5\(\hat{j}\) + 4\(\hat{k}\)) + 11 = 0, and passing through the point (-2, 1, 3).

Answer 1

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

2.(2 + 3λ) + 1.(-7-5λ) + 3.(4 + 4λ) = 3-11λ 

-4-6λ-7-5λ + 12 + 12λ-3 + 11λ = 0 

-14 + 12 + 12λ = 0

λ = 1/6

Putting the value of λ in equation (2)