volume of parallelepiped w/ adjacent edges

**rcmango** · February 23rd 2009, 11:29 PM

P(2, 0, -1), Q(4, 1, 0), R(3, -1, 1), S(2, -2, 2)

must find the volume of the parallelepiped with adjacent edges PQ PR and PS.

I tried setting it up and i still didn't come out with the correct answer of 3

...whats confusing me is the amount of points above, there are 4 now, instead of 2, or 3.

please help.

**Soroban** · February 24th 2009, 05:02 AM

Hello, rcmango!

Did you work out the vectors?

$P(2, 0, \text{-}1),\quad Q(4, 1, 0),\quad R(3,\text{-}1, 1),\quad S(2,\text{-}2, 2)$

Find the volume of the parallelepiped with adjacent edges $PQ, PR, PS.$

We have: . $\begin{array}{ccccc}<br /> \vec u &=& \overrightarrow{PQ} &=& \langle 2,1,1\rangle \\ \vec v &=& \overrightarrow{PR} &=& \langle1,\text{-}1,2\rangle \\ \vec w &=& \overrightarrow{PS} &=& \langle 0,\text{-}2,3\rangle \end{array}$

Then: . $\text{Volume} \;=\;\bigg|\vec u \cdot(\vec v \times \vec w)\bigg|$

. . Hence: . $V \;=\;\left|\,\begin{bmatrix}2&1&1\\1&\text{-}1&2 \\ 0&\text{-}2&3\end{bmatrix}\,\right|\quad\hdots$ etc.

**rcmango** · February 24th 2009, 10:45 AM

okay i see,

after i calculated PQ PR and PS

ex: PQ (4-2) + (1-0) + (0+1) 2, 1, 1 ........... and the others....

i used their cross product for:

i came out with 2*(2-(-4) - 1(3-0) + 1(-2 - 0)

everything looks fine except i got a -3,
thats very close though, probably just another wrong negative somewhere i did.

thanks for the help.