# Thread: Proving Cross product is orthogonal

1. ## Proving Cross product is orthogonal

Let u,v in R3. Define the cross product of u and v to be the vector
u x v=(det [u2 u3 / v2 v3], det [u3 u1 / v3 v1], det [u1 u2/ v1 v2])
Show that u x v is orthogonal to u and v.

I started by taking the dot product of u with u x v and v with u x v, and showing that that needs to be zero. However, I'm having a hard time applying the dot product when I simplify it, and I think I'm just getting stuck on something fairly silly.
I get
u dot(u2v3-u3v2), u dot(u1v3-u3v1), u dot(u1v2-u2v1)
And I'mn ot sure how to proceed.

2. ## Re: Proving Cross product is orthogonal

$\displaystyle (u_1,u_2,u_3) \cdot (u_2v_3-u_3v_2,u_3v_1-u_1v_3,u_1v_2-u_2v_1)$

$\displaystyle = u_1(u_2v_3-u_3v_2) + u_2(u_3v_1-u_1v_3) + u_3(u_1v_2-u_2v_1)$

$\displaystyle = u_1u_2v_3 - u_1u_3v_2 + u_2u_3v_1 - u_1u_2v_3 + u_1u_3v_2 - u_2u_3v_1$

$\displaystyle = u_1u_2(v_3 - v_3) + u_2u_3(v_1 - v_1) + u_1u_3(v_2 - v_2)$

$\displaystyle = 0 + 0 + 0 = 0$

$\displaystyle v \cdot (u \times v)$ is calculated similarly.

3. ## Re: Proving Cross product is orthogonal

Originally Posted by renolovexoxo
Let u,v in R3. Define the cross product of u and v to be the vector
u x v=(det [u2 u3 / v2 v3], det [u3 u1 / v3 v1], det [u1 u2/ v1 v2])
Show that u x v is orthogonal to u and v.
For simplicity let $\displaystyle u=<a,b,c>~\&~v=<d,e,f>$ then $\displaystyle u\times v=<bf-ce,~cd-af,~ae-bd>$
Now do the dot products.

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