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Dado tres vectores <img src="http://latex.codecogs.com/gif.latex?\overrightarrow{x},\overrightarrow{y},\overrightarrow{z}" title="\overrightarrow{x},\overrightarrow{y},\overrightarrow{z}" align="absmiddle" /></font><font bgcolor="#FFFFFF" size="4"><font color="#000000">, su producto mixto [</font></font><font size="4"><img src="http://latex.codecogs.com/gif.latex?\overrightarrow{x},\overrightarrow{y},\overrightarrow{z}" title="\overrightarrow{x},\overrightarrow{y},\overrightarrow{z}" align="absmiddle" />] es el NÚMERO que resulta de efectuar:
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<font size="4"><img src="http://latex.codecogs.com/gif.latex?[\overrightarrow{x},\overrightarrow{y},\overrightarrow{z}]=\overrightarrow{x}*(\overrightarrow{y}\wedge%20\overrightarrow{z})" title="[\overrightarrow{x},\overrightarrow{y},\overrightarrow{z}]=\overrightarrow{x}*(\overrightarrow{y}\wedge \overrightarrow{z})" /></font>
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SI: <img src="http://latex.codecogs.com/gif.latex?\left.\begin{matrix}%20\overrightarrow{x}%20&(x_1,&x_2%20,&x_3)%20\\%20\overrightarrow{y}%20&(y_1,%20&y_2%20,&y_3)%20\\%20\overrightarrow{z}%20&(z_1,%20&z_2%20,%20&z_3)%20\end{matrix}\right\}" title="\left.\begin{matrix} \overrightarrow{x} &(x_1,&x_2 ,&x_3) \\ \overrightarrow{y} &(y_1, &y_2 ,&y_3) \\ \overrightarrow{z} &(z_1, &z_2 , &z_3) \end{matrix}\right\}" align="absmiddle" /> <img src="http://latex.codecogs.com/gif.latex?%5CRightarrow%20%5Coverrightarrow%7By%7D%5Cwedge%20%5Coverrightarrow%7Bz%7D=%28%5Cbegin%7Bvmatrix%7D%20y_2%20&y_3%20%5C%5C%20z_2%20&z_3%20%5Cend%7Bvmatrix%7D,%5Cbegin%7Bvmatrix%7D%20y_1%20&y_3%20%5C%5C%20z_1%20&z_3%20%5Cend%7Bvmatrix%7D,%5Cbegin%7Bvmatrix%7D%20y_1%20&y_2%5C%5C%20z_1%20&z_2%20%5Cend%7Bvmatrix%7D%29" title="\Rightarrow \overrightarrow{y}\wedge \overrightarrow{z}=(\begin{vmatrix} y_2 &y_3 \\ z_2 &z_3 \end{vmatrix},\begin{vmatrix} y_1 &y_3 \\ z_1 &z_3 \end{vmatrix},\begin{vmatrix} y_1 &y_2\\ z_1 &z_2 \end{vmatrix})" height="47" width="315" align="absmiddle" /> LUEGO:<img src="http://latex.codecogs.com/gif.latex?%5Coverrightarrow%7Bx%7D*%20%28%5Coverrightarrow%7By%7D%5Cwedge%20%5Coverrightarrow%7Bz%7D%29=%20x_1*%20%5Cbegin%7Bvmatrix%7D%20y_2%20&y_3%20%5C%5C%20z_2%20&%20z_3%20%5Cend%7Bvmatrix%7D-%20x_2%20*%20%5Cbegin%7Bvmatrix%7D%20y_1%20&y_3%20%5C%5C%20z_1%20&z_3%20%5Cend%7Bvmatrix%7D%20+%20x_3%20%5Cbegin%7Bvmatrix%7D%20y_1%20&y_2%20%5C%5C%20z_1%20&z_2%20%5Cend%7Bvmatrix%7D%20=%20%5Cbegin%7Bvmatrix%7D%20x_1%20&%20x_2%20&x_3%20%5C%5C%20y_1%20&y_2%20&y_3%20%5C%5C%20z_1%20&%20z_2%20&z_3%20%5Cend%7Bvmatrix%7D" title="\overrightarrow{x}* (\overrightarrow{y}\wedge \overrightarrow{z})= x_1* \begin{vmatrix} y_2 &y_3 \\ z_2 & z_3 \end{vmatrix}- x_2 * \begin{vmatrix} y_1 &y_3 \\ z_1 &z_3 \end{vmatrix} + x_3 \begin{vmatrix} y_1 &y_2 \\ z_1 &z_2 \end{vmatrix} = \begin{vmatrix} x_1 & x_2 &x_3 \\ y_1 &y_2 &y_3 \\ z_1 & z_2 &z_3 \end{vmatrix}" height="59" width="485" /></font>
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PROPIEDADES</b></u></font>
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Son las mismas que la de los determinantes:
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<b>A-</b> <img src="http://latex.codecogs.com/gif.latex?[%5Coverrightarrow%7Bx%7D,%5Coverrightarrow%7By%7D,%5Coverrightarrow%7Bz%7D]=%20-[%5Coverrightarrow%7Bx%7D,%5Coverrightarrow%7By%7D,%5Coverrightarrow%7Bz%7D]=[%5Coverrightarrow%7Bx%7D,%5Coverrightarrow%7By%7D,%5Coverrightarrow%7Bz%7D]=%20-[%5Coverrightarrow%7Bx%7D,%5Coverrightarrow%7By%7D,%5Coverrightarrow%7Bz%7D]=..." title="[\overrightarrow{x},\overrightarrow{y},\overrightarrow{z}]= -[\overrightarrow{x},\overrightarrow{y},\overrightarrow{z}]=[\overrightarrow{x},\overrightarrow{y},\overrightarrow{z}]= -[\overrightarrow{x},\overrightarrow{y},\overrightarrow{z}]=..." /></font>
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<b>B-</b> <img src="http://latex.codecogs.com/gif.latex?[%5Calpha%20%5Coverrightarrow%7Bx%7D,%5Cbeta%20%5Coverrightarrow%7By%7D,%5Cgamma%20%5Coverrightarrow%7Bz%7D]=%5Calpha%20*%20%5Cbeta%20*%5Cgamma%20*[%5Coverrightarrow%7Bx%7D,%5Coverrightarrow%7By%7D,%5Coverrightarrow%7Bz%7D]" title="[\alpha \overrightarrow{x},\beta \overrightarrow{y},\gamma
\overrightarrow{z}]=\alpha * \beta *\gamma
*[\overrightarrow{x},\overrightarrow{y},\overrightarrow{z}]" /></font>
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<b>C-</b> <img src="http://latex.codecogs.com/gif.latex?[\overrightarrow{x}+\overrightarrow{x}',\overrightarrow{y},\overrightarrow{z}]=[\overrightarrow{x},\overrightarrow{y},\overrightarrow{z}]+[\overrightarrow{x}',\overrightarrow{y},\overrightarrow{z}]" title="[\overrightarrow{x}+\overrightarrow{x}',\overrightarrow{y},\overrightarrow{z}]=[\overrightarrow{x},\overrightarrow{y},\overrightarrow{z}]+[\overrightarrow{x}',\overrightarrow{y},\overrightarrow{z}]" /></font>
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<b>D-</b> Si <img src="http://latex.codecogs.com/gif.latex?\overrightarrow{x},\overrightarrow{y},\overrightarrow{z}" title="\overrightarrow{x},\overrightarrow{y},\overrightarrow{z}" /> son COPLANARIOS <img src="http://latex.codecogs.com/gif.latex?%5CRightarrow%20[%5Coverrightarrow%7Bx%7D,%5Coverrightarrow%7By%7D,%5Coverrightarrow%7Bz%7D]=0" title="\Rightarrow [\overrightarrow{x},\overrightarrow{y},\overrightarrow{z}]=0" /></font>
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