Basis for Plane Perpendicular to a Line

[veritaserum20]

Homework Statement

Let L be the line in R^3 spanned by v1=(1,1,1)

Find a basis (v2,v3) for the plane perpendicular to L, and verify that B=(v1,v2,v3) is a basis for R^3.

Homework Equations

The Attempt at a Solution

I know that if two vectors are perpendicular or orthogonal that their dot product is equal to zero. However, I am not sure how to find a plane that is perpendicular to a vector.

 

[LCKurtz]

Hint: If you cross v1 with anything not parallel to it you will get a vector v2 perpendicular to v1. What direction will v2 cross v1 have?

 

[vela]

You could also solve the equation

$$\begin{pmatrix}1 & 1 & 1\end{pmatrix}\begin{pmatrix}x \\ y \\ z\end{pmatrix} = 0$$

You'll get a two-dimensional solution from which you can read off two basis vectors.