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I have a problem in vector algebra. In this Wolfram-page the last two formulas (9) and (10) are: $$ \frac{ | (x_2 - x_1) \times (x_1 - x_0) | }{|x_2 - x_1 |} = \frac{ | (x_0 - x_1) \times (x_0 - x_2) | }{|x_2 - x_1 |} $$ I've tried to apply properties founded in this other Wolfram-crossproduct-page but I still don't understand it. How is possible? And why the do this vectors manipulation?

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$\begin{align}|(x_2-x_1)\times(x_1-x_0)|&=|(x_1-x_2)\times(x_0-x_1)|\\&=|(x_0-x_1)\times(x_1-x_2)|\\&=|(x_0-x_1)\times((x_0-x_1)+(x_1-x_2))|\\&=|(x_0-x_1)\times(x_0-x_2)|\end{align}$

where the second is because you are in magnitude signs and the third is because the cross product of a vector with itself (or any parallel vector) is zero.

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  • $\begingroup$ absolute value?? i thought the $|v \times w|$ was the magnitude of the cross product between v and w is magnitude not absolute value.. $\endgroup$
    – nkint
    Jan 12, 2011 at 23:09
  • $\begingroup$ You are right it is magnitude, but the magnitude is always positive. I should have said it that way and will fix. I just wanted to justify that changing the order of the factors did not change the result, whereas for the cross product itself it changes the sign. $\endgroup$ Jan 12, 2011 at 23:28
  • $\begingroup$ ok now it's more clear. but the remaining question is.. why they do that manipulation? $\endgroup$
    – nkint
    Jan 13, 2011 at 10:59
  • $\begingroup$ @nkint: Just to give you more options for using the result, I think. $\endgroup$ Jan 13, 2011 at 13:38

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