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Given a line through space defined by

$$l(t) = a + tb$$

and a normal direction $\vec{n}$, write out a parametric equation for the plane containing the line and perpendicular to the normal. Also, when will the plane be undefined?

I know that we can write the equation of a plane given the normal vector and a point on the plane, but the equation above doesn't really give us this. What is the proper way to do this?

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1 Answer 1

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$b$ is the direction of the line.

That means that $(b \times n)$ is the direction of another line passing through the plane (crossproduct produces a perpendicular vector).

That means that $l_2(t) = a + s (b \times n)$ is another line passing through the plane.

$l$ and $l_2$ allow you to choose 3 non-colinear points that are in the plane (for example, $s = t = 0$, $s = 1$, $t=1$). From 3 points you should be able to find the plane equation.

Think about what relationship $b$ and $n$ must have for $l_2$ to not represent a line (it will degenerate to a single point instead).

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