Help find the equation of two planes

Question

I have the question

Consider the line L through the distinct points A = (a,b,c) and D = (d,e,f) Find the equations of the two planes which intersect at right angles along L

MY ATTEMPTED SOLUTION

I can find the equation of the line L by

D-A = (d,e,f) - (a,b,c)

    = (d-a, e-b, f-c)

This gives the directional vector

choosing a starting point p1 = (a,b,c)

Now to find the parametric equation r = (a,b,c) + t(d-a, e-b, f-c) this gives individual parametric equations

x = a + t(d-a)

y = b + t(e-b)

z = c + t(f-c)

How do I progress? am I on the right track? thanks for the help!

Answer 1

Finding a direction vector is some progress. Parametric equations are not needed.

Using a point on the line, a plane will be determined by its normal vector. This normal vector must be orthogonal to the direction vector of the line. There are many such vectors; in fact, the solution to your problem is far from unique.

The tool you need is the cross product. Pick any vector $v$ that is not collinear to the direction vector $DA$. The cross-product $DA\times v$ gives you a normal vector $n$ for the first plane. Then $DA\times n$ gives a normal vector to the second plane, since it's orthogonal to both $DA $ and $n$.