[Math] Finding an orthogonal projection matrix onto the plane

Question

Find the orthogonal projection matrix onto the plane $$x + y – z = 0$$

The solution to this video recitation video on MIT open courseware immediately states that we can chose

$$a_1 = \begin{pmatrix}
1\\
-1\\
0
\end{pmatrix} \ \ \ \ \ \ \ \ \text{and} \ \ \ \ \ \ \ a_2 = \begin{pmatrix}
1\\
0\\
1
\end{pmatrix}$$
So then $$A = \begin{pmatrix}
1 & 1\\
-1 & 0 \\
0 & 1
\end{pmatrix}$$
and we simply apply the equation $P = A(A^{T}A)^{-1}A^{T}$. The lecturer did not explain how he can choose such an $a_1$ and $a_2$. Can anyone explain that?

Answer 1

The lecturer simply chose two vectors $a_{1}$ and $a_{2}$ that are independent and contained in the plane $x+y-z=0$. He then applied the formula that you mentioned.