Vector Cross-Multiplication Operation

A Vector Cross-Multiplication Operation is a binary vector operation in three-dimensional space that is a product operation.

• AKA: ×, Vector Cross Product.
• Context:
  • It can be calculated as [math]\displaystyle{ \vec{a} \times \vec{b} \equiv \begin{pmatrix}a_x\\a_y\\a_z\end{pmatrix} \times \begin{pmatrix}b_x\\b_y\\b_z\end{pmatrix} = \begin{pmatrix}a_2b_3-a_3b_2\\a_3b_1-a_1b_3\\a_1b_2-a_2b_1\end{pmatrix} }[/math]
• Example(s):
  • [math]\displaystyle{ (1,2,3) \times (7,6,5) \equiv (-8, 15, -8) }[/math].
  • [math]\displaystyle{ (1,2,3) \times (2,4,6) \equiv (0, 0, 0) }[/math].
  • [math]\displaystyle{ (1,2,3) \times (-1,-2,-3) \equiv (-4, 2, 0) }[/math].
  • …
• Counter-Example(s):
  • a Dot Product.
  • a Scalar Product.
• See: Product, Proportionality Equation, Euclidean Space, Perpendicular, Normal (Geometry), Algebra Over a Field, Commutative.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/cross_product Retrieved:2015-1-17.
  • In mathematics, the cross product or vector product is a binary operation on two vectors in three-dimensional space and is denoted by the symbol ×. The cross product a × b' of two linearly independent vectors a and b is a vector that is perpendicular to both and therefore normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming.

    If two vectors have the same direction (or have the exact opposite direction from one another, ie. are not linearly independent) or if either one has zero length, then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular, for perpendicular vectors, this is a rectangle and the magnitude of the product is the product of their lengths. The cross product is anticommutative (i.e. ) and is distributive over addition (i.e.). The space and product form an algebra over a field, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket.

    Like the dot product, it depends on the metric of Euclidean space, but unlike the dot product, it also depends on a choice of orientation or "handedness". The product can be generalized in various ways; it can be made independent of orientation by changing the result to pseudovector, or in arbitrary dimensions the exterior product of vectors can be used with a bivector or two-form result. Also, using the orientation and metric structure just as for the traditional 3-dimensional cross product, one can in n dimensions take the product of vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions. If one adds the further requirement that the product be uniquely defined, then only the 3-dimensional cross product qualifies. (See Generalizations, below, for other dimensions.)

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/multiplication_of_vectors Retrieved:2015-2-7.
  • … Cross product — also known as the “vector product", a binary operation on two vectors that results in another vector. The cross product of two vectors in 3-space is defined as the vector perpendicular to the plane determined by the two vectors whose magnitude is the product of the magnitudes of the two vectors and the sine of the angle between the two vectors. So, if n is the unit vector perpendicular to the plane determined by vectors A and B,

    A × B = ||A|| ||B|| sin θ n.

• http://en.wikipedia.org/wiki/Cross_product#Coordinate_notation
  • Using column vectors, we can represent the same result as follows:

[math]\displaystyle{ \begin{pmatrix}s_1\\s_2\\s_3\end{pmatrix}=\begin{pmatrix}u_2v_3-u_3v_2\\u_3v_1-u_1v_3\\u_1v_2-u_2v_1\end{pmatrix} }[/math]

• http://en.wikipedia.org/wiki/Cross_product#Generalizations
  • There are several ways to generalize the cross product to the higher dimensions.
    • 9.1 Lie algebra.
    • 9.2 Quaternions.
    • 9.3 Octonions.
    • 9.4 Wedge product.
    • 9.5 Multilinear algebra

2009

• (WordNet, 2009) ⇒ http://wordnetweb.princeton.edu/perl/webwn?s=cross%20product
  • S: (n) vector product, cross product (a vector that is the product of two other vectors)

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