Find the orthogonal projection of y onto u.
-$$\mathbf { y } = \left[ \begin{array} { r }
- 12 \
36
\end{array} \right] , \mathbf { u } = \left[ \begin{array} { r }
- 8 \
4
\end{array} \right]$$
A)
$\left[ \begin{array} { r } - 24 \ 12 \end{array} \right]$

B)
$\left[ \begin{array} { r } - 8 \ 4 \end{array} \right]$

C)
$\left[ \begin{array} { r } - 24 \ 4 \end{array} \right]$

D)

$\left[ \begin{array} { l } - \frac { 8 } { 3 } \ - \frac { 4 } { 3 } \end{array} \right]$

Question

Find the orthogonal projection of y onto u.
-$$\mathbf { y } = \left[ \begin{array} { r } 
- 12 \
36
\end{array} \right] , \mathbf { u } = \left[ \begin{array} { r } 
- 8 \
4
\end{array} \right]$$
A)
$\left[ \begin{array} { r } - 24 \ 12 \end{array} \right]$

B)
$\left[ \begin{array} { r } - 8 \ 4 \end{array} \right]$

C)
$\left[ \begin{array} { r } - 24 \ 4 \end{array} \right]$

D)

$\left[ \begin{array} { l } - \frac { 8 } { 3 } \ - \frac { 4 } { 3 } \end{array} \right]$

Quizplus · Accepted Answer

The orthogonal projection of $\mathbf{y}$ onto $\mathbf{u}$ is given by the formula $\text{proj}_{\mathbf{u}}(\mathbf{y}) = \frac{\mathbf{y} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u}$. Calculating the dot products, $\mathbf{y} \cdot \mathbf{u} = (-12)(-8) + (36)(4) = 96 + 144 = 240$ and $\mathbf{u} \cdot \mathbf{u} = (-8)(-8) + (4)(4) = 64 + 16 = 80$. Thus, $\text{proj}_{\mathbf{u}}(\mathbf{y}) = \frac{240}{80} \mathbf{u} = 3 \mathbf{u} = 3 \left[ \begin{array} { r } - 8 \ 4 \end{array} \right] = \left[ \begin{array} { r } - 24 \ 12 \end{array} \right]$.

Find the Orthogonal Projection of Y onto U y=[−1236],u=[−84]\mathbf { y } = \left[ \begin{array} { r } - 12 \\36\end{array} \right] , \mathbf { u } = \left[ \begin{array} { r } - 8 \\4\end{array} \right]y=[−1236​],u=[−84​]

Find the Orthogonal Projection of Y onto U $\mathbf { y } = \left[ \begin{array} { r } - 12 \\36\end{array} \right] , \mathbf { u } = \left[ \begin{array} { r } - 8 \\4\end{array} \right]$