I was asked to find an orthonormal basis for the plane $x + 2y +3z =0$.
I found a regular basis, $(-2,1,0),(-3,0,1)$, and then performed the Gram-Schmidt process to find 2 orthonormal vectors that span the plane. I got the vectors $\frac{1}{\sqrt{5}}(-2,1,0), \frac{1}{\sqrt{70}}(-3,-6,5)$.
Then, I was asked to complete the basis to a basis of $\mathbb{R}^3$. So, I am looking for a vector $(x,y,z)$ such that $(x,y,z)\cdot\frac{1}{\sqrt{5}}(-2,1,0) = 0,\ (x,y,z)\cdot\frac{1}{\sqrt{70}}(-3,-6,5) = 0,\ x^2 + y^2 + z^2 = 1$.
From the first 2 equations, I made a system, and after reducing I got the matrix
$$
\begin{pmatrix}
-2 & 1 & 0\\
-3 & 0 & 1
\end{pmatrix}
$$
After looking, I realized that the rows of this matrix are the original basis I found for the plane.
Does this happen for a reason or is it just a coincidence?
Best Answer
What you notice is not a coincidence. It is just a consequence of the way the dot product is expressed using the vector coordinates.
And your last vector is the cross product of the previous two. Which avoids the computation you mention.