So I am given two points P and Q and a vector $\vec v$ in $R^3$. I want to find the plane that goes through P and Q and is parallel to v. So I made a vector PQ, but how can I make that into plane parallel to $\vec v$?
[Math] Plane passing through 2 points, parallel to a vector
linear algebra
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Best Answer
Generate a second $\vec{PQ}$ vector from P and Q. Take the cross product $\vec{PQ} \times \vec{v}$. Call it $\vec{N}$, since it is a normal vector to the plane. This is because both $\vec{PQ}$ and $\vec{v}$ are parallel to the plane.
Note that a plane is a collection of vectors which are perpendicular to a normal and which passes through a point. So we want any vector: $\vec{s} = \langle x-P_x, y-P_y, z-P_z\rangle$ to be perpendicular to n.
What does it mean for two vectors to be perpendicular? Their dot product is zero. So...
$\vec{N} \cdot \vec{s} =$
$N_x(x-P_x) + N_y(y-P_y) + N_z(z-P_z) = 0$