Question

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

basis of R³

Answer 1

Denote $u=(2,0,3)$. A vector $v=(v_1,v_2,v_3)\perp u$ if

$$u.v=0\implies2v_1+3v_3=0\implies v_3=-\tfrac{2}{3}v_1\;(\text{and } v_2 \text{ is arbitrary}).$$

Take, for example $v_1=3,\;v_3=-2\text{ and }v_2=0$. Then $v=(3,0,-2)$ is orthogonal to $u$.

To construct a third vector $w=(w_1,w_2,w_3)$ orthogonal to $u\text{ and }v$ we follow the same technique: $u.w\text{ and }v.w=0$ give

$$\begin{cases}2w_1+3w_3=0\\3w_1-2w_3=0\end{cases}\implies w_1=w_3=0\text{ and }w_2\text{ is arbitrary}$$

Take, for example, $w=(0,1,0)$.

Then $\{(2,0,3),(-3,0,2),(0,1,0)\}$ is an orthogonal basis of $\mathbb{R}^3$.