The planes vector r.(2i + λj - 3k) = 10 and vector r.(λi + 3j + k) =

1.	The planes vector r.(2i + λj - 3k) = 10 and vector r.(λi + 3j + k) = 5 are perpendicular. Find λ.
Answer» Since the planes are perpendicular, the angle between the normals = 90°. The normals are vector n₁ = vector(2i + λj - 3k) and vector n₂ = vector(λi + 3j + k) vector(n₁.n₂) = 0 [∵ θ = π/2] ⇒ (2)(λ) + (λ)(3) + (-3)(1) = 0 ⇒ 2λ + 3λ – 3 = 0 5λ – 3 = 0 ⇒ 5λ = 3 ⇒ λ = 3/5

The planes vector r.(2i + λj - 3k) = 10 and vector r.(λi + 3j + k) = 5 are perpendicular. Find λ.

Answer»

Since the planes are perpendicular, the angle between the normals = 90°.

The normals are vector n₁ = vector(2i + λj - 3k) and vector n₂ = vector(λi + 3j + k)

vector(n₁.n₂) = 0 [∵ θ = π/2]

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

5λ – 3 = 0 ⇒ 5λ = 3 ⇒ λ = 3/5