Proving Vector Equality Using Cross Products

[TheBigBadBen]

This is a question from Yahoo! answers.

The Question:

Suppose that we know vectors v ×u= w×u for every three-dimensional vector w . Prove that v = w.
Hint: Choose a few simple vectors
for u and compute the cross products. What does this say about the coordinates of v and
w?

 

[TheBigBadBen]

My Answer:

So, we have two vectors $\vec v=(v_1,v_2,v_3)$ and $\vec w=(w_1,w_2,w_3)$ such that
$$\vec v\times \vec u=\vec w \times \vec u$$
for every choice of vector $\vec u$. We can say that the two vectors are the same if $v_1=w_1$, $v_2=w_2$, and $v_3=w_3$. As the hint suggests, I will choose the following vectors to put in the place of $\vec u$: $\vec u_1 = (1,0,0)$ and $\vec u_2 = (0,1,0)$. We know that
$$\vec v\times \vec u_1=\vec w \times \vec u_1 \\
\vec v\times \vec u_2=\vec w \times \vec u_2$$
Finding these products gives us the following set of equations:
$$
(0,v_3,-v_2)=(0,w_3,-w_2) \\
(-v_3,0,v_1)=(-w_3,0,w_1)
$$
Setting each of the coordinates equal, the first equation tells us that $v_3=w_3$ and $-v_2=-w_2$. The second equation tells us that, because the third entries are equal, $v_1=w_1$. With these three pieces of information, we can now say that $\vec v = \vec w$.