equation of a plane through 2 points and parallel to a line

Question

what is the equation of a plane passing through 2 given points (p 1) and (p 2) and parallel to a given line L 1? i know how to find the equation of a plane passing through a point with position vector a and parallel to 2 lines with vectors b and c which is given by: r=a+n(b)+m(c) any answer with the formula and how you derived it(i.e. the thought process as well) would be much appreciated!

Answer 1

In reply to comments:

Well, there's a few different ways to define a plane. There's the way you will know which is a plane is defined by a point and two vectors in the plane $\mathbf r = \mathbf a + \lambda \mathbf b + \mu \mathbf c$. That's all you need to express everything we need to know about the plane in order to define it uniquely, but there's a way of simplifying this using the cross-product (if you haven't done the cross product, its an operation which combines two vectors to generate a third vector that is perpendicular to both).

If we take the cross product of the two lines in the plane, we get a (nearly) unique vector that's perpendicular to the plane. I say nearly because the vector could point the other way. Anyway, since this vector is perpendicular to the plane, we can say that the dot product of our vector with any vector in the plane is zero. So this gives us another way of expressing a plane. If $\mathbf r$ is the position vector of a general point in the plane, $\mathbf a$ is the position vector of a specific point in the plane, and $\mathbf v$ is a vector perpendicular to the plane, then:

$$ (\mathbf r - \mathbf a)\cdot \mathbf v = 0$$

We can link this back to our original equation by noting that $\mathbf v = \mathbf b \times \mathbf c$ is a vector perpendicular to the plane.

There's one final way of defining a plane - in Cartesian coordinates. It doesn't involve vectors though, and you'll probably come across it soon if you haven't already, so I won't explain it.