82ba58…: “Instead of the amazing owl”

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Consider the planes defined by the equations below.

$\begin {array}{lr} P_1: \quad 2x-y+3z &=4 \\ P_2: \quad 4x-2y+6z &= 5 \\ P_3: \quad 5x+2y+z &=7 \\ \end {array}$

The planes $P_1$ and $P_2$ are parallelorthogonalneither parallel nor orthogonal .

The planes $P_1$ and $P_3$ are parallelorthogonalneither parallel nor orthogonal .

The planes $P_2$ and $P_3$ are parallelorthogonalneither parallel nor orthogonal .

Two planes are parallel if and only if their normal vectors are parallel, and orthogonal if and only if their normal vectors are orthogonal.

A normal vector to $P_1$ is $\vec {n}_1 = \vector {2,-1,3}$ and a normal vector to $P_2$ is $\vec {n}_2 = \vector {4,-2,6}$ .

Note that $\vec {n}_1 \dotp \vec {n}_2 = \answer {28} \neq 0$ , so $P_1$ and $P_2$ are not orthogonal.

To check if these plane are parallel, we can check if the normal vectors are parallel. We can do this by either checking if $\vec {n}_1 \cross \vec {n}_2 = \vec {0}$ or by checking if $\vec {n}_1$ is a nonzero scalar multiple of $\vec {n}_2$ . Which is easier to check?