I need to find the volume of a parallelpiped. The volume is spanned by 3 vectors $$\begin{cases}a=(-5,-3,2), \\ b=(1,0,2), \\ c=(2,-1,3), \end{cases}$$ so I tried with the formula $(a \times b) \cdot c$, to find $a \times b$ I got $$\begin{bmatrix} -3 & 2 \\ 0 & 2 \end{bmatrix}, - \begin{bmatrix} -5 & 2 \\ 1 & 2 \end{bmatrix}, \begin{bmatrix} -5 & -3 \\ 1 & 0 \end{bmatrix},$$ and then I calculate this and it gives me $-6-0 , - (-10-2) , (-5 \cdot 0)-1\ \cdot (-3)$ and that's $$\begin{bmatrix}-6 & 12 & 3 \end{bmatrix}^t \cdot \begin{bmatrix}2 & -1 & 3 \end{bmatrix},$$ this is just $c$ :). I get $-12-12+9=-15$ so basically my volume is negative? I don't know if that's the way its supposed to be or if I have done anything wrong, any tips/solutions? :)
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$\begingroup$ Your answer is correct. The negative sign is accounting for a orientation change. When you compute volumes using determinants you compute an oriented volume, which in your case means your three vectors are not compatible with the right hand rule. $\endgroup$– Mark FantiniApr 8, 2014 at 3:55
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$\begingroup$ Do you know the right hand rule? $\endgroup$– Mark FantiniApr 8, 2014 at 4:05
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$\begingroup$ Okay, so we have a starting point. The right hand rule tells you the sign convention: whenever you can align vectors following it the orientation is positive. If you compute $a \times b$ using the right hand rule you will get an opposite vector regarding the rule, hence the negative. This is an orientation reversal. $\endgroup$– Mark FantiniApr 8, 2014 at 4:08
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$\begingroup$ Yes, the volume magnitude is 15. You should probably explain somewhere about the orientation. $\endgroup$– Mark FantiniApr 8, 2014 at 4:14
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$\begingroup$ The volume gets multiplied by corresponding factors. If you magnify one vector it gets magnified by that factor, if you magnify all vectors then it gets magnified by the product of the factors. In the first case it gets magnified by $k$, in the second by $k^3$. Compute the explicit example for $k=2$ to see how this happens. =) $\endgroup$– Mark FantiniApr 8, 2014 at 4:19
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The volume is determined by the absolute value of (a×b)⋅c. Also, you can create a 3x3 matrix with your given vectors then find it's determinant (which turns out to be the same operations you've already performed); and once again the volume is determined by the absolute value of the determinant.
