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So I understand most of the properties of cross products. However I ran into a small complication.

I get that $i\times j = k$, $j\times k = i$. I also understand that $k \times j = -i$ and that $k\times k = j\times j = i\times i = 0$.

But what happens when you have $-k\times k$? Does that also equal $0$, or something else?

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Well, it is $-0$. And $-0=0$. – Giuseppe Negro Sep 16 '12 at 3:15
up vote 2 down vote accepted

There is another algebraic property of the cross product such that $(ra) \times b = r(a \times b)$ where $a, b$ are vectors and $r$ is a constant. (

Since you said that $k \times k = 0$, we can write $(-k) \times k = -(k \times k) = -0 = 0$

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