As one of the requirements for my course, we were given a task to answer which is determining the cross product of the given: U = (1, -4, 5, 7) V =(1, 2, 6, 7)
I've already searched tons of forums and most of them answer that this is not possible. Yet I wish to provide solid proof and arguments as I present my answer to my professor. Your help would be very much appreciated. Thank you!
Note that both of your vectors have $7$ as the forth component. So you can `ignore' the fourth component for a moment and find the cross product of $(1,-4,5)$ and $v=(1,2,6)$ which for that order is $(-10,5,6)$. So if for the cross product you want a vector perpendicular to both $u,v$, you can take $(-10,5,6,0)$. This is a vector is certainly perpendicular to the plane spanned by $u,v$.
If instead of $7$ the fourth component was $0$, we could find infinitely many $4$ tuples orthogonal to $u,v$ unless we impose some extra condition.
So the question then is what the person who wrote the question meant by this exactly.