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Next: Curves in the plane Up: Analytic geometry in the Previous: Cross product.

The box product.

Definition 1.24   Let $\overrightarrow{u} $, $\overrightarrow{v} $ and $\overrightarrow{w} $ be three vectors in the 3-dimensional space. Their triple scalar product is the number

\begin{displaymath}( \overrightarrow{u}\times \overrightarrow{v} ) \cdot \overrightarrow{w}\end{displaymath}

Remark 1.25   By Def.  1.6 and Def.  1.17, we have:

\begin{displaymath}\vert ( \overrightarrow{u}\times \overrightarrow{v} ) \cdot \...
...rrow{v} \vert \cdot \vert \overrightarrow{w} \vert \cos \theta
\end{displaymath}

where $\theta$ is the angle between the vector $\overrightarrow{w} $ and a vector normal to the plane of $\overrightarrow{u} $ and $\overrightarrow{v} $ (cf Fig.  8).


  
Figure 8: The parallelepiped on three given vectors.
\begin{figure}
\mbox{\epsfig{file=BoxProduct.eps,height=4cm} }
\end{figure}

Thus, the absolute value of the triple scalar product is equal the volume of the box whose sides are given by the vectors $\overrightarrow{u} $, $\overrightarrow{v} $ and $\overrightarrow{w} $.

Proposition 1.26 (Determinant Formula)       

For any three vectors $\overrightarrow{u} $, $\overrightarrow{v} $ and $\overrightarrow{w} $, we have:

\begin{displaymath}( \overrightarrow{u}\times \overrightarrow{v} ) \cdot \overri...
...& u_2 & u_3 \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \end{vmatrix}\end{displaymath}

where $\overrightarrow{u} =(u_1,u_2.u_3)$, $\overrightarrow{v} =(v_1,v_2.v_3)$ and $\overrightarrow{w} =(w_1,w_2.w_3)$with respect to an orthonormal direct basis.

Example 1.27   Let $\overrightarrow{u} =(1,2,3)$, $\overrightarrow{v} =(2,3,1)$ and $\overrightarrow{w} =(0,1,2)$. We have:

\begin{displaymath}( \overrightarrow{u}\times \overrightarrow{v} ) \cdot \overri...
...rix} + 2 \begin{vmatrix}1 & 2 \\ 2 & 3 \end{vmatrix}= 5 -2 =3.
\end{displaymath}

Thus, the volume of the parallelepiped whose sides are given by $\overrightarrow{u} $, $\overrightarrow{v} $ and $\overrightarrow{w} $ is equal to 3.


next up previous contents
Next: Curves in the plane Up: Analytic geometry in the Previous: Cross product.
Noah Dana-Picard
2001-05-30