Cross product: Difference between revisions

Latest revision as of 03:59, 5 January 2008

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Vector product

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-{{subpages}}
+#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.
-[[Category:CZ Live]]
-[[Category:Physics Workgroup]]
-[[Category:Mathematics Workgroup]]

Cross product: Difference between revisions

Latest revision as of 03:59, 5 January 2008

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