Volume of parallelopiped

**SyNtHeSiS** · October 12th 2010, 10:47 AM

What is the volume of a parallelopiped with adjacent vertices at O(0, 0, 0), A(1, 1, -2), B(3, 1, -1) and C(2, -1, 1)?

I got the vectors:

OA = (1,1,-2)

OB = (3,1,-1)

OC = (2,-1,1)

but I am not sure which 2 vectors I must cross (and obviously the other one will get dotted). How do you know which 2 vectors are in the same plane in order to cross them?

**HallsofIvy** · October 12th 2010, 11:10 AM

It doesn't matter! $\vec{u}\cdot(\vec{v}\times\vec{w})$ and the other 5 permutations of the vectors differ only in sign. Since $volume= \left|\vec{u}\cdot(\vec{v}\times\vec{w})\right|$ the sign does not matter.

In fact, with $\vec{u}= x_1\vec{i}+ \y_1\vec{j}+ z_1\vec{3}$ , $\vec{v}= x_2\vec{i}+ \y_2\vec{j}+ z_2\vec{k}$ , and $\vec{w}= x_3\vec{i}+ y_3\vec{j}+ z_3\vec{3}$ , since $\vec{v}\times\vec{w})= \left|\begin{array}{ccc}\vec{i} & \vec{j} & \vec{k} \\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3\end{array}\right|$ , it is easy to see that $\vec{u}\cdot(\vec{v}\times\vec{w})= \left|\begin{array}{ccc}x_1 & y_1 & z_ 1\\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3\end{array}\right|$ .

Permuting the vectors just permutes the rows which only changes the sign. And, again, the volume is the absolute value of that.

(Three points determine a plane. Any two vectors, having the same "initial point", lie in a plane.)

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