So I have
and I must find an orthogonal non zero vector.
If I find the cross product it should be an orthogonal vector, right? So how do I find a non zero vector?
Here is my work:
So can the vector 5,-1,-4 be considered the non zero vector orthogonal to points PQR?
You don't really need to take the dot product since you know that cross product always produces a vector orthogonal to both- you only need the cross product to check if you did the cross product correctly.
"So can the vector 5,-1,-4 be considered the non zero vector orthogonal to points PQR?"
Well, it is a vector perpendicular to the two vectors, not "the" vector- there are an infinite number of vectors orthogonal to two given vectors.
If you are simply asking if this vector is "non zero", of course it is! The only vector that is NOT "non zero" is the zero vector (0, 0, 0).
By the way, both here and in your title you talk of vectors "orthogonal to points". A vector is NEVER orthogonal to any points- "orthogonal" or "perpendicular" is only defined for lines and vectors, not points.