This is an old revision of the document!
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/syntax/header.php on line 56
Warning: Undefined array key "do" in /home/levene/public_html/w/mst10030/lib/plugins/revealjs/action.php on line 14
The area of a parallelogram
Consider a parallelogram, two of whose sides are $\def\vv{\vec v}\def\ww{\vec w}\vv$ and $\ww$.
This has double the area of the triangle considered above, so its area is $\|\vv\times\ww\|$.
Example
A triangle with two sides $\def\c#1#2#3{\begin{bmatrix}#1\\#2\\#3\end{bmatrix}}\vv=\c13{-1}$ and $\ww=\c21{-2}$ has area $\tfrac12\|\vv\times\ww\|=\tfrac12\left\|\c13{-1}\times\c21{-2}\right\|=\tfrac12\left\|\c{-5}0{-5}\right\|=\tfrac52\left\|\c{-1}0{-1}\right\|=\tfrac52\sqrt2$, and the parallelogram with sides $\vv$ and $\ww$ has area $\|\vv\times\ww\|=5\sqrt2$.
The volume of a parallelepiped in $\mathbb R^3$
Let $\def\uu{\vec u}\uu$, $\vv$ and $\ww$ be vectors in $\bR^3$.
Consider a parallelepiped, with three sides given by $\uu$, $\vv$ and $\ww$.
Call the face with sides $\vv$ and $\ww$ the base of the parallelpiped. The area of the base is $A=\|\vv\times\ww\|$, and the volume of the parallelpiped is $Ah$ where $h$ is the height, measured at right-angles to the base.
One vector which is at right-angles to the base is $\vv\times\ww$. It follows that $h$ is the length of $\vec p=\text{proj}_{\vv\times\ww}\uu$, so \[ h=\|\text{proj}_{\vv\times\ww}\uu\|=\left\|\frac{\uu\cdot(\vv\times\ww)}{\|\vv\times\ww\|^2}\vv\times\ww\right\| = \frac{\uu\cdot(\vv\times\ww)}{\|\vv\times\ww\|^2}\|\vv\times\ww\| = \frac{|\uu\cdot(\vv\times\ww)|}{\|\vv\times\ww\|}\] so the volume is \[ V=Ah=\|\vv\times\ww\|\frac{|\uu\cdot(\vv\times\ww)|}{\|\vv\times\ww\|}\] or \[ V=|\uu\cdot(\vv\times\ww)|,\] so $V$ is the absolute value of the determinant $\begin{vmatrix}u_1&u_2&u_3\\v_1&v_2&v_3\\w_1& w_2&w_3\end{vmatrix}$: \[ V=\left|\quad \begin{vmatrix}u_1&u_2&u_3\\v_1&v_2&v_3\\w_1& w_2&w_3\end{vmatrix}\quad \right|.\]
Example
Find volume of a parallelepiped whose vertices include $A=(1,1,1)$, $B=(2,1,3)$, $C=(0,2,2)$ and $D=(3,4,1)$, where $A$ is an adjacent vertex to $B$, $C$ and $D$.
Solution
The vectors $\vec{AB}=\c102$, $\vec{AC}=\c{-1}11$ and $\vec{AD}=\c230$ are all edges of this parallepiped, so the volume is \[ V=\left|\quad \begin{vmatrix}1&0&2\\-1&1&1\\2&3&0\end{vmatrix}\quad \right| = | 1(0-3)-0+2(-3-2)| = |-13| = 13.\]