Cross Product

The result of the cross-product of a two vectors is another vector. Given two vectors $\vec{a}=<a_{1}, a_{2}, a_{3}>$ and $\vec{b}=<b_{1}, b_{2}, b_{3}>$, their cross-product can be written as:

\begin{eqnarray*}
\vec{a} \times \vec{b} &=& <a_{2}b_{3}-a_{3}b_{2}, a_{3}b_{1}-a_{1}b_{3}, a_{1}b_{2}-a_{2}b_{1}> 
\end{eqnarray*}

or, using determinants:

\begin{eqnarray*}
\vec{a} \times \vec{b} &=& \left| \begin{matrix} 
a_{2} & a_{3} \\
b_{2} & b_{3}
\end{matrix} \right| i - \left| \begin{matrix} 
a_{1} & a_{3} \\
b_{1} & b_{3}
\end{matrix} \right| j - \left| \begin{matrix} 
a_{1} & a_{2} \\
b_{1} & b_{2}
\end{matrix} \right| k
\end{eqnarray*}

which is derived from:

\begin{eqnarray*}
\vec{a} \times \vec{b} &=& \left| \begin{matrix} 
i & j & k \\
a_{1} & a_{2} & a_{2} \\
b_{1} & b_{2} & b_{3}
\end{matrix} \right|
\end{eqnarray*}

Properties

  • The length of the resulting vector representing the cross-product of two vectors is:

\begin{eqnarray*}
|\vec{a} \times \vec{b}|&=&|\vec{a}||\vec{b}|\sin{\theta}
\end{eqnarray*}

where the vertical bars represent the scalar value of the vector.

  • The length of the cross-product of two vectors is equal to the area of the parallelogram determined by the two vectors.



\begin{tikzpicture}
  % grid
  \draw[help lines] (-3,-2) grid (4,2);
  
  % origin
  %\draw[red, line width=.1mm] (-0.1,-0.1) -- (0.1,0.1)
  %  (0.1,-0.1) -- (-0.1,0.1);
  %\coordinate[label={[red]above:$O$}] (O) at (0,0);
  
  %\draw[black,line width=5mm] (-3, {3 * sqrt(3) - sqrt(3)}) -- ((-3,0);
  %\node [square,rotate={30},minimum size=10mm] at (-3, {3 * sqrt(3) - sqrt(3)}) [draw] (d2) [orange,fill,text=white] {$d_{2}$};
  %\draw[orange,line width=.1mm] (-3, {3 * sqrt(3) - sqrt(3)}) -- (-2,0); 
  %\draw[orange,line width=.1mm] (-3, {3 * sqrt(3) - sqrt(3)}) -- (0,2);
  
  % coordinates
  \coordinate[label={[black]left:$A$}] (A) at (-1,1.5);
  \coordinate[label={[black]left:$B$}] (B) at (-2,-1);
  \coordinate[label={[black]right:$C$}] (C) at (2,-1);
  \coordinate[label={[black]right:$D$}] (D) at (3,1.5);
  \coordinate[label={[black]below:$P$}] (P) at (-1,-1);
  
  % mark P
  \drawpoint{P}{.5mm}{black}
  
  % theta 
  \markangle{B}{A}{C}{2mm}{2mm}{$\theta$}{green}{south west}
  
  % triangle 
  \draw[black, line width=.1mm] (A) -- (B) -- (C) -- (D) -- cycle;
  
  % perpendicular
  \draw[black, line width=.1mm] (A) -- (P);
  
  % alpha 
  \node [square,minimum size=1mm,dotted] at (-1.14,-0.85) [draw] (d2) [black] {};
  
  % braces
  %\drawbrace{B}{C}{2mm}{blue}{$a$}{0}{-4mm}{mirror}
  %\drawbrace{A}{P}{2mm}{red}{$h$}{4mm}{0}{}
  %\drawbrace{A}{B}{2mm}{green}{$c$}{-4mm}{0}{mirror}
  %\drawbrace{A}{C}{2mm}{red}{$b$}{3mm}{3mm}{}
  \drawbrace{A}{P}{2mm}{black}{${\color{cyan}|\vec{b}|}\sin{\color{green}{\theta}}$}{10mm}{0mm}{}
  \drawbrace{B}{C}{2mm}{blue}{$|\vec{a}|$}{0}{-4mm}{mirror}
  \drawbrace{A}{B}{2mm}{cyan}{$|\vec{b}|$}{-6mm}{2mm}{mirror}
  
\end{tikzpicture}

\begin{eqnarray*}
A_{\blacksquare} &=&  {\color{blue}|\vec{a}|}{\color{cyan}|\vec{b}|}\sin{\color{green}{\theta}}
\end{eqnarray*}

  • Multiplication by scalars is commutative. For example, suppose that $c$ is a scalar and that $\vec{a} \times \vec{b}$ is the cross product of $\vec{a}$ respectively $\vec{b}$, in that case we have that:

\begin{eqnarray*}
c(\vec{a}) \times \vec{b} &=& \vec{a} \times c(\vec{b})
\end{eqnarray*}

  • The cross-product is distributive.

\begin{eqnarray*}
\vec{a} \times (\vec{b} + \vec{c}) &=& \vec{a} \times \vec{b} + \vec{a} \times \vec{c}
\end{eqnarray*}

  • The volume of the parallelepiped determined by the vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$ is equal to the magnitude of their scalar triple product.
  • The vector triple product of the vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$ is given by:

\begin{eqnarray*}
\vec{a} \times (\vec{b} \times \vec{c}) &=& (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}
\end{eqnarray*}

  • Two vectors are parallel if and only if their cross product is the zero vector.

Determining Triangle Area in Three Dimensions

One solution to find the area of a triangle is to determine its height and then proceed to the multiply the height by the base and divide by two. However, that may be a difficult task and instead, by using the parallelogram property of vectors, one can calculate the area of the parallelogram and then divide the area by two.


fuss/mathematics/geometry/vectors.txt ยท Last modified: 2017/02/22 18:30 (external edit)

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