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Let P be the matrix that projects onto j that is, we want P to satisfy

\(\mathbf{P} \mathbf{v} = \text{The projection of $\mathbf{v}$ onto } \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \) for all vectors v.

Use the pictures below to calculate \(\mathbf{P}\mathbf{i}, \mathbf{P} \mathbf{j}, \mathbf{P}\mathbf{k} \).

 

Calculate the matrix P that projects onto j.

 Aug 15, 2019
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\(\text{$v$ projected onto $(0,1,0)$ is $(v \cdot (0,1,0))(0,1,0)$}\\ v \cdot (0,1,0) = v_y\\ v \perp (0,1,0) = (0,v_y,0)\\ \text{The matrix that will perform this is}\\ P=\begin{pmatrix}0&0&0\\0&1&0\\0&0&0\end{pmatrix}\)

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 Aug 16, 2019

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