I have a set with two vectors in $R^3$. What method should I use to find the vectors orthogonal to both in the original set?

Question

There are so many different methods I've found on SE and through Matlab, and they're all giving me different results.

Specifically, I have {v1} = (1,2,1) and {v2} = (2,1,0) in set S. What is the method to find {v3} vectors that are orthogonal to both v1 and v2?

I'm preparing for a final and I'm trying to find a flexible method for many cases. The answer I got for above was v3 = {1,-2,3} but different methods are returning different results.

Answer 1

Guide:

Your answer should be a non-zero scalar multiple of the $v_3$ that you provided since $v_1$ and $v_2$ are not parallel to each other.

Method $1$:

• Compute the cross product of $v_1$ and $v_2$, that will give you a valid solution.

Method $2$:

• Solve the linear system $v_1^Tx=0$ and $v_2^Tx=0$ by reducing the system say to REF. You will obtain multiple solution, of which all of them are scalar multiple of $v_3$.

Either method should be fine.

Answer 2

What I would do:

• Compute the planes orthogonal to both of your vectors.

A plane orthogonal to the vector $(a,b,c)$ has the equation $ax + by + cz + d = 0, \forall d \in \mathbb{R}$

• Compute the intersection of the two planes by replacing the first plane equation in the second one