Show that the following planes are at right angles: vector r.(2i - j +

1.	Show that the following planes are at right angles: vector r.(2i - j + k) = 15 and vector r.(i - j - 3k) = 3
Answer» The normal vectors are vector n₁ = vector(2i - j + k) and vector n₂ = vector(i - j - 3k) vector(n₁.n₂) = (2)(1) + (-1)(-1) + (1)(-3) = 2 + 1 – 3 = 0 ⇒ vector(n₁ ⊥r to n₂), i.e., the normals to the planes are at right angles. So, the planes are at right angles.

Show that the following planes are at right angles: vector r.(2i - j + k) = 15 and vector r.(i - j - 3k) = 3

Answer»

The normal vectors are

vector n₁ = vector(2i - j + k) and vector n₂ = vector(i - j - 3k)

vector(n₁.n₂) = (2)(1) + (-1)(-1) + (1)(-3) = 2 + 1 – 3 = 0

⇒ vector(n₁ ⊥r to n₂), i.e., the normals to the planes are at right angles. So, the planes are at right angles.