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:

\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}> 

or, using determinants:

\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

which is derived from:

\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|


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

|\vec{a} \times \vec{b}|&=&|\vec{a}||\vec{b}|\sin{\theta}

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.

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

  • 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:

c(\vec{a}) \times \vec{b} &=& \vec{a} \times c(\vec{b})

  • The cross-product is distributive.

\vec{a} \times (\vec{b} + \vec{c}) &=& \vec{a} \times \vec{b} + \vec{a} \times \vec{c}

  • 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:

\vec{a} \times (\vec{b} \times \vec{c}) &=& (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}

  • 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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