Question

Find the orthogonal projection of onto the subspace of spanned by and.

Short answer

werrer

Step-by-step solution

Determine the orthonormal basis

Consider a set spanned and a vector .

If the vectoris perpendicularthen.

By the theorem of orthogonal basis, the orthogonal is defined as follows.

Substitute the value for in the equation as follows.

$\begin{array}{r}{\overrightarrow{u}}_1=\frac{{\overrightarrow{v}}_1}{∥{\overrightarrow{v}}_1∥}\\ {\overrightarrow{u}}_1=\frac{\left[\begin{array}{c}2\\ 3\\ 6\end{array}\right]}{∥\left[\begin{array}{c}2\\ 3\\ 6\end{array}\right]∥}{\overrightarrow{u}}_1=\frac{1}{\sqrt{2^2+3^2+6^2}}\left[\begin{array}{c}3\\ 6\end{array}\right]\\ {\overrightarrow{u}}_1=\frac{1}{7}\left[\begin{array}{c}2\\ 3\\ 6\end{array}\right]\end{array}$