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Suppose I have a vector equation:

\begin{cases}

x=0+10t\\

y=0+10t\\

z=0+10t

\end{cases}

Which forms the symmetric equation [itex]\frac{x-0}{10}=\frac{y-0}{10}=\frac{z-0}{10}[/itex]

Now, I know the symmetric equations can be split up so that you can form the two planes whose intersection yields the initial vector:

[itex]\frac{x-0}{10}=\frac{y-0}{10}[/itex] and [itex]\frac{y-0}{10}=\frac{z-0}{10}[/itex]

but I haven't been able to find any examples on how to get from the split symmetric equations of the line to two separate equations of a plane in standard form.

Would I just cross multiply each?

and get

Plane 1: x = y or x - y + 0z = 0

Plane 2: y = z or 0x + y - z = 0

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# Equations of Planes from Symmetric Equation of a Line

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