Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Mathematics
General Math
What is the Definition and Properties of a Cross Product in Vector Algebra?
Reply to thread
Message
[QUOTE="Greg Bernhardt, post: 4804026, member: 1"] [SIZE="4"][U][B]Definition/Summary[/B][/U][/SIZE] The cross product of two vectors [itex]\mathbf{A}[/itex] and [itex]\mathbf{B}[/itex] is a third vector (strictly, a pseudovector or axial vector) [itex]\mathbf{A}\times\mathbf{B}[/itex] perpendicular to both of the original vectors, with magnitude equal to the product of their magnitudes times the (positive) sine of the angle between them, and in the direction determined by the right-hand rule. It is anti-commutative: [itex]\mathbf{A}\times\mathbf{B}\,=\,-\,\mathbf{B}\times\mathbf{A}[/itex] If the vectors [itex]\mathbf{A}[/itex] and [itex]\mathbf{B}[/itex] are considered as 1-forms, then the wedge product [itex]\mathbf{A}\wedge\mathbf{B}[/itex] is a 2-form (a directed area), and its dual [itex]\ast(\mathbf{A}\wedge\mathbf{B})[/itex] is a dual 1-form, corresponding to the pseudovector [itex]\mathbf{A}\times\mathbf{B}[/itex] [SIZE="4"][U][B]Equations[/B][/U][/SIZE] The magnitude of [itex]\mathbf{A}\times\mathbf{B}[/itex] is the area of the paralleogram with sides [itex]\mathbf{A}[/itex] and [itex]\mathbf{B}[/itex]: [tex]|\mathbf{A}\times\mathbf{B}| = AB \sin \theta[/tex] In terms of Cartesian components: [tex](\mathbf{A}\times\mathbf{B})_x = A_y B_z - A_z B_y[/tex] [tex](\mathbf{A}\times\mathbf{B})_y = A_z B_x - A_x B_z[/tex] [tex](\mathbf{A}\times\mathbf{B})_z = A_x B_y - A_y B_x[/tex] The triple product (scalar product) [itex]\mathbf{A}\cdot(\mathbf{B}\times\mathbf{C})[/itex] is the volume of the parallepiped with sides [itex]\mathbf{A}[/itex] [itex]\mathbf{B}[/itex] and [itex]\mathbf{C}[/itex], and therefore: [tex]\mathbf{A}\cdot(\mathbf{B}\times\mathbf{C})\,=\,\mathbf{B}\cdot(\mathbf{C}\times\mathbf{A})\,=\,\mathbf{C}\cdot(\mathbf{A}\times\mathbf{B})[/tex] The repeated cross product [itex]\mathbf{A}\times(\mathbf{B}\times\mathbf{C})[/itex] is a vector perpendicular to [itex]\mathbf{A}[/itex], and is a linear combination of [itex]\mathbf{B}[/itex] and [itex]\mathbf{C}[/itex]: [tex]\mathbf{A}\times(\mathbf{B}\times\mathbf{C})\,=\,(\mathbf{A}\cdot\mathbf{C})\mathbf{B}\,-\,(\mathbf{A}\cdot\mathbf{B})\mathbf{C}[/tex] [SIZE="4"][U][B]Extended explanation[/B][/U][/SIZE] For two vectors [itex]\mathbf{a} = \left(a_x,a_y,a_z\right)\;\;\mathbf{b}=\left(b_x,b_y,b_z\right)[/itex] in [itex]\mathbb{R}^3[/itex], the cross poduct can be written as the determinant of a 3x3 matrix: [tex]\mathbf{a}\times\mathbf{b} = \left|\begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z\end{array}\right|[/tex] Where [itex]\mathbf{i}, \mathbf{j}, \mathbf{k}[/itex] is a right-handed orthonormal basis. [B]Polar vectors and pseudovectors:[/B] A polar (ordinary) vector is reversed under an inversion of the coordinate axes: [itex]\mathbf{a}\ \mapsto\ -\mathbf{a}[/itex] However, the cross product of two polar vectors is [I]not[/I] reversed: [itex]\mathbf{a}\ \times\ \mathbf{b}\ \mapsto\ (-\mathbf{a})\ \times\ (-\mathbf{b})\ =\ \mathbf{a}\ \times\ \mathbf{b}[/itex] In other words, the cross product of two polar (ordinary) vectors is invariant (the same) under an inversion of the coordinate axes: this is called a [COLOR="Red"]pseudovector[/COLOR] (axial vector). [B]Directed area:[/B] A [COLOR="Red"]directed area [/COLOR]is a flat surface together with a magnitude (its area), and a sign ([itex]\pm[/itex]) indicating a direction of rotation within the surface (alternatively, indicating a preferred normal direction). A directed area is an elementary [COLOR="red"]2-form[/COLOR] (a [COLOR="red"]wedge product[/COLOR] of two [COLOR="Red"]1-forms[/COLOR]). [SIZE="1"]A [I]general[/I] 2-form is a [I]sum[/I] of directed areas.[/SIZE] Its dual (in three-dimensional space) is a 1-form in the [COLOR="Red"]dual space[/COLOR], corresponding to a [COLOR="red"]pseudovector[/COLOR] normal to the surface and with magnitude equal to its area. [B]Triple product 3-form pseudoscalars and directed volume:[/B] The wedge product of [I]three [/I]1-forms is a [COLOR="Red"]3-form.[/COLOR] In three-dimensional space, all 3-forms are multiples of each other, and so a 3-form is essentially a scalar. To be precise, a 3-form is the dual of a [COLOR="red"]0-form[/COLOR], which [I]is[/I] a scalar. However, under an inversion of the coordinate axes, a 3-form is multiplied by minus-one, and so a 3-form technically is a [COLOR="Red"]psuedoscalar[/COLOR]. The[COLOR="red"] triple product[/COLOR] [itex]\mathbf{a}\cdot(\mathbf{b}\,\times\,\mathbf{c})[/itex] of three vectors is the dot product of a vector and a pseudovector, and is the pseudoscalar equal to the wedge product [itex]\mathbf{a}\,\wedge\,\mathbf{b}\,\wedge\,\mathbf{c}[/itex] of the three vectors, [I]in the same order[/I]. It is the [COLOR="red"]directed volume[/COLOR] of the parallepiped whose sides are those three vectors ("directed" because it includes a sign ([itex]\pm[/itex]) indicating a direction of rotation around the common vertex). * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks! [/QUOTE]
Insert quotes…
Post reply
Forums
Mathematics
General Math
What is the Definition and Properties of a Cross Product in Vector Algebra?
Back
Top