Question

Complete { (2, 0, 3)} to form an orthogonal

basis of R³

Answer 1

Let us complete $\{ (2, 0, 3)\}$ to form an orthogonal basis. Consider the vector $(3,0,-2).$ Since $2\cdot 3+0\cdot0+3\cdot(-2)=0,$ we conclude that the inner product of $(2, 0, 3)$ and $(3,0,-2)$ is zero, and hence the vectors $(2, 0, 3)$ and $(3,0,-2)$ are orthogonal. Further, consider the vector $(0,1,0)$. It follows that $2\cdot 0+0\cdot 1+3\cdot 0=0$ and $3\cdot 0+0\cdot 1-2\cdot 0=0,$ and hence the vector $(0,1,0)$ is orthogonal to the vectors $(2, 0, 3)$ and $(3,0,-2).$ Therefore, $\{ (2, 0, 3),(3,0,-2),(0,1,0)\}$ is an orthogonal basis of $\R^3.$