if a, b, c, and r are any vector, prove [a.bxc]r = (a.r)bxc + (b.r)cxa + (c.r)axb

Question

Im using a period as dot product and x for cross product. I've been stuck on this problem for a long time. My main question is, if there is a vector next to a parenthesis does that imply cross or dot product?

Answer 1

I will answer your "main question".  Note that each of the 3 parentheses is a dot product (i.e. a scalar), so each parenthesis is a scalar multiplying the vector expression to the right.  In the bracket, you have a vector dotted into a cross product, so that too is a scalar, multiplying the vector to the right.  Note that

 

a·bxc = a·(bxc) = scalar

 

It cannot equal (a·b)xc because (a·b) is a scalar, and you cannot have the cross product of a scalar with a vector.  Note also that (a·r), (b·r), and (c·r) are scalars.  Since these are scalars multiplying a vector, the multiplication is neither a dot or cross product.