Exercise 6

This subsection shows how to project orthogonally in two ways, the method of Example 3.2 and 3.3, and the method of Theorem 3.8. To compare them, consider the plane $P$ specified by $3x+2y-z=0$ in $\mathbb{R}^{3}$.

• Find a basis for $P$.

• Find $P^{\perp}$ and a basis for $P^{\perp}$.

• Represent this vector with respect to the concatenation of the two bases from the prior item.

  $$\vec{v}=\begin{pmatrix}1\\ 1\\ 2\end{pmatrix}$$

• Find the orthogonal projection of $\vec{v}$ into $P$ by keeping only the $P$ part from the prior item.

• Check that against the result from applying Theorem 3.8.