The question asked by Olugbenga here is quite elementary. I am answering it in full because he may not be the only one struggling with this concept even though I have explained it on several occasions in class and in my notes. It is about the fundamental issue of a product of a symmetric quantity and an anti-symmetric one. If you don’t fully grasp this, you may get some assistance from the explanation provided here:
Further to the above proof of the vanishing of this triple products, there are other ways to explain why this quantity must vanish. To an SS3 child preparing for JAMB, the argument could be that we consider the cross product in the parenthesis first. Recall that the cross product of two vectors produces a vector that is perpendicular to both vectors. It is therefore in a plane perpendicular to the one containing both vectors. u, being one of these vectors must have a zero dot product with this cross product.
To a 300-le3vel undergraduate, a geometrical argument could be constructed from the fact that a triple product can be seen as the volume of the parallelepiped formed by the three vectors. If it turns out that two of the vectors are actually the same vector, the volume of such a parallelepiped is obviously zero.
The tensor argument developed above is of course requiring a more advanced development of tensors, summation convention and contraction.
Thank you sir. What a learning forum! Am okay now.
“It is a principle that the product of a symmetric quantity with an anti-symmetric quantity in the same indices will give a result of zero”.
Actually, i have solved the question and got the expressed answer but couldn’t tend the expression to zero. So i made guessed assumption that in that case it will be zero from perpendicularity concept of normal vector.
But now i have more reasons and fact for my argument.
THANK YOU EVERY,EVERY MUCH SIR.
There are several other silent issues in the above proof. If you take a closer look, you will see that it also actually proofs that changing the dot and the cross in a triple product will NOT alter its value! It proofs this in the rather elementary symmetry of all its vectors such that it does not matter which had a dot or which had a cross at the beginning! Furthermore, it also proves that a swap of the cross product will alter sign because this would cause an alteration in the alternating tensor and therefore change sign!
Tensor analysis makes vectors look like primary one work!
Thank you very much sir, for shedding more light on the question raised by Gbenga.