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[Community Question] Linear-algebra: Write a vectorial equation which is perpendicular to a plane and goes through a point.

One of our user asked:

Consider plane $\beta: -x+y=1$ and the point $A=(0;0;2)$. I'm asked to write a vectorial equation for the line perpendicular to plane $\beta$ which also goes through point $A$.

I start by identifying a normal vector to $\beta$, which is $\vec{r}=(-1;1;0)$. This vector must be the "director vector" (not sure of the correct term in english) to the line I want.

Hence the equation for the line is $\left(x;\:y;\:z\right)=\left(0;0;2\right)+k\left(-1;1;0\right),\:k\:\in \mathbb{R}$.

However, when I plot it, with various values for $k$, I get this:

enter image description here

My questions are: is this right? And if so - where exactly is the perpendicular line?


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