# Thread: Proving a property of the cross/dot products?

1. ## Proving a property of the cross/dot products?

1. 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?

2. ## Re: Proving a property of the cross/dot products?

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.)

3. ## Re: Proving a property of the cross/dot products?

I am told to post the entire question online so I did. But the BOLD part is the part I am struggling with. Can you read the question please. Thank you.

4. ## Re: Proving a property of the cross/dot products?

OK. Well, here is a hint:

The dot product of two vectors is zero if they are orthogonal (perpendicular in the general sense), the cross product is zero if they are parallel.

5. Originally Posted by kandygirl16
1. 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?
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.

6. ## Re: Proving a property of the cross/dot products?

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

7. ## Re: Proving a property of the cross/dot products?

What do you mean "the cross product of two vectors is perpendicular to both." ?

8. ## Re: Proving a property of the cross/dot products?

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