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Consider the following planes: \(\begin{align*} &P_1 \text{ is the plane consisting of all points $(x, y, z)$ such that $x+y + 2z = 1$}, \\ &P_2 \text{ is the plane consisting of all points $Q$ such that }\overrightarrow{OQ} = \begin{pmatrix}1 \\ -1\\ 4 \end{pmatrix} + s \begin{pmatrix} 1 \\ 2 \\ 5 \end{pmatrix} + t \begin{pmatrix} 0 \\ -3 \\ 4 \end{pmatrix} \text{for some real $s$ and $t$},\\ &P_3 \text{ is the plane consisting of all points $(x, y, z)$ such that $3x+ 5y + 7z = 0$}, \\ &P_4 \text{ is the plane consisting of all points $Q$ such that }\overrightarrow{OQ} = \begin{pmatrix}1 \\ -2\\ 5 \end{pmatrix} + s \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix} + t \begin{pmatrix} 1 \\ 0 \\ 3 \end{pmatrix} \text{for some real $s$ and $t$}. \end{align*}\)

For each plane above, figure out whether it goes through the origin, and answer "yes" or "no" in the order above. (Answer with "yes" if it does go through the origin.)

 

I don't know how to see if it goes to the origin or not, can someone please help? Ty in advance!

 Mar 2, 2020
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The answers are yes, no, no, yes.

 Mar 3, 2020

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