- Prove that u DOT (v CROSS w)= ( u CROSS v) DOT w . Can you give criterion for when this product is zero, based on the directions of the three vectors?
A straight forward way is to just prove it directly using computation.
Let $\displaystyle \vec u = \left< u_1,u_2,u_3 \right>$, $\displaystyle \vec v = \left< v_1, v_2, v_3 \right>$ and $\displaystyle \vec w = \left< w_1,w_2,w_3\right>$.
Compute each side of the equation separately and show that they actually give you the same thing. (We know to pick 3D vectors here since we have a cross product. Cross products are only defined for 3D vectors.)
Sorry this is just an exercise in subscripts.
$\displaystyle <u_1,u_2,u_3>\cdot(<v_1,v_2,v_3>\times<w_1,w_2,w_3 >)$
$\displaystyle <u_1,u_2,u_3>\cdot(<v_2w_3-v_3w_1,v_3w_1-v_1w_3,v_1w_3-v_3w_1>)$
$\displaystyle u_1(v_2w_3-v_3w_1)+u_2(v_3w_1-v_1w_3)+u_3(v_1w_3-v_3w_1)$
Now sort it all out.
Jhevon has the right idea for figuring out when the product is zero. You'll also need to know that the cross product of two vectors is perpendicular to both.
Your answer is going to be pretty simple - it's not like "u is perpendicular to v and not w, or v is perpendicular to u and not w, or ..."
- Hollywood
If the two vectors aren't in the same direction or opposite directions (the cross product is zero in this case), you can find a line that's perpendicular to both, and that's the direction of the cross product.
The article on Wikipedia, Cross product - Wikipedia, the free encyclopedia, has some pictures that might help you see it.
- Hollywood