+0

# Consider the following planes:

-1
45
1
+82

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.)