Find the vector equation of the plane passing through the intersection

1.	Find the vector equation of the plane passing through the intersection of the planes and through the point (2,1,3).
Answer» Let the planes be P₁ : 2x + 2y -3z – 7 = 0, P₂ : 2x +5y + 3z = 9 ∴ Required equation be P₁+ λ P₂ = 0 (2x + 2y – 3z – 7) + λ(2x + 5y + 3z – 9) = 0, passes through (2, 1, 3) (4 + 2 – 9 – 7) + λ(4 + 5 + 9 – 9) = 0 - 10 + λ9 = 0 ⇒ λ = \(\frac{10}{9}\) Required plane is (2x + 2y – 3z – 7) + λ(2x + 5y + 3z – 9) = 0 18x + 18y -27z – 63 + 20x + 50y + 50z – 90 = 0 38x + 68y + 3z – 153 = 0

Find the vector equation of the plane passing through the intersection of the planes and through the point (2,1,3).

Answer»

Let the planes be

P₁ : 2x + 2y -3z – 7 = 0,

P₂ : 2x +5y + 3z = 9

∴ Required equation be P₁+ λ P₂ = 0

(2x + 2y – 3z – 7) + λ(2x + 5y + 3z – 9) = 0,

passes through

(2, 1, 3) (4 + 2 – 9 – 7) + λ(4 + 5 + 9 – 9) = 0

- 10 + λ9 = 0 ⇒ λ = \(\frac{10}{9}\)

Required plane is

(2x + 2y – 3z – 7) + λ(2x + 5y + 3z – 9) = 0

18x + 18y -27z – 63 + 20x + 50y + 50z – 90 = 0

38x + 68y + 3z – 153 = 0