Let and Find a vector which is perpendicular to both and and

Given, and

We need to find a vector perpendicular to and such that.

Recall a vector that is perpendicular to two vectors and is

Here, we have (a₁, a₂, a₃) = (1, 4, 2) and (b₁, b₂, b₃) = (3, –2, 7)

So, is a vector parallel to.

Let for some scalar λ.

We have.

⇒ λ[(2)(32) + (–1)( –1) + (4)(–14)] = 15

⇒ λ(64 + 1 – 56) = 15

⇒ 9λ = 15

So, we have.

Thus,