Cross product: Difference between revisions

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#REDIRECT [[vector product]]
 
The cross product, or vector product, is a type of [[vector space|vector]] multiplication in <math>\mathbb{R}^3</math>, and is widely used in many areas of mathematics and physics. In general Euclidean spaces there is another type of multiplication called the [[dot product]] ( or scalar product). Both the dot product and the cross product are widely used in in the study of optics, mechanics, electromagnetism, and gravitational fields, for example. 
 
=== Definition ===
Given two vectors, <b>A</b> = (A<sub>x</sub>,A<sub>y</sub>,A<sub>z</sub>) and <b>B</b> = (B<sub>x</sub>,B<sub>y</sub>,B<sub>z</sub>) in <math>\mathbb{R}^3</math>, the cross product is defined as the vector product of the magnitude of <b>A</b>, the magnitude of <b>B</b>, the sine of the smaller angle between them, and a unit vector (<b>a<sub>N</sub></b>) that is perpendicular (or normal to) the plane containing vectors <b>A</b> and <b>B</b> and which follows the right-hand rule (see below). 
 
<b>A</b> <b>x</b> <b>B</b> = <b>a<sub>N</sub></b> |<b>A</b>||<b>B</b>|sinθ<sub>AB</sub>
 
 
where <math>|\mathbf A|=(\mathbf A \cdot \mathbf A)^{1/2}</math> and <math>|\mathbf B|=(\mathbf B \cdot \mathbf B)^{1/2}</math> are, respectively, the magnitudes of <b>A</b> and <b>B</b>. See [[dot product]] for the evaluation of this equation.
 
 
Reversing the order of the vectors <b>A</b> and <b>B</b> results in a unit vector in the opposite direction, meaning that the cross product is not commutative, and thus:
 
 
<b>B</b> <b>x</b> <b>A</b> = -(<b>A</b> <b>x</b> <b>B</b>)
 
The cross product of any vector with itself (or another parallel vector) is zero because the sin(0) = 0.
 
<b>A</b> <b>x</b> <b>A</b> = 0
 
 
=== Another formulation ===
 
Rather than determining the angle and perpendicular unit vector to solve the cross product,  the form below is often used to solve the cross product in <math>\mathbb{R}^3</math>.
 
 
<b>A</b> <b>x</b> <b>B</b> = (A<sub>y</sub>B<sub>z</sub> - A<sub>z</sub>B<sub>y</sub>)<b>a</b><sub>x</sub> + (A<sub>z</sub>B<sub>x</sub> - A<sub>x</sub>B<sub>z</sub>)<b>a</b><sub>y</sub> + (A<sub>x</sub>B<sub>y</sub> - A<sub>y</sub>B<sub>x</sub>)<b>a</b><sub>z</sub>,
 
where <b>a</b><sub>x</sub>, <b>a</b><sub>y</sub> and <b>a</b><sub>z</sub> are the orthonormal bases on which <b>A</b> and <b>B</b> have been defined.  The above formula can be written more concisely in the following form:
 
<math>
\mathbf A \times \mathbf B =
\left|\begin{array}{ccc} \mathbf a_x & \mathbf a_y & \mathbf a_z \\
A_x & A_y & A_z \\
B_x & B_y & B_z \end{array} \right|,
</math>
 
where <math>\left|\cdot\right|</math> denotes the determinant of a matrix.
 
 
== The right hand rule ==
The diagram below illustrates the direction of <b>A x B</b>, which follows the right hand rule.
If one points the fingers of the right hand towards the head of vector <b>A</b> (with the wrist at the origin), then curls them towards the direction of <b>B</b>, the extended thumb will point in the direction of <b>A x B</b>.
 
[[Image:CrossProduct.jpg|center|frame|Vector Diagram illustrating the direction of <b>A x B</b>]]
 
 
 
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Latest revision as of 03:59, 5 January 2008

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