Vector product proof...?

Question

I'm asked to prove that:

grad x grad * scalar = 0

But I can't seem to get it to work out. Nothing seems to cancel out in my determinant. Anybody have any suggestions or ideas where I might be going wrong? Maybe a website that could help?

Its hard for me to type out the entire determinant I have here since there is no ASCII symbol for the partial derivative.

Answer 1

The first thing you should note is that the fact that the vector is a gradient is irrelevant: EVERY vector has a cross product of zero with any scalar multiple of itself. That should simplify your calculations some. So simply prove this for an arbitrary vector.

Also, I would not multiply out the formula. Instead, I would use the properties of determinants and cross product. Specifically:

a·b×c = det([a, b, c]), where [a, b, c] denotes the matrix with vectors a, b, and c respectively (since the determinant of every matrix is equal to the determinant of its transpose, it doesn't really matter whether they are column or row vectors). Now let a and b be any vectors and k any scalar. Then [a, b, kb] is singular (it has an obvious linear dependency), so det([a, b, kb]) = 0, for all vectors a. Of course, the only vector with the property that a·v=0 for all vectors a is the zero vector, so v=b×kb=0. Q.E.D.