The Cross Product - Revision history

Khanh: /* Coordinate notation */

2022-01-20T19:57:32Z

Coordinate notation

Khanh: /* Infinitesimal generators of rotations */

2022-01-20T19:56:11Z

Infinitesimal generators of rotations

Khanh: /* Lorentz force */

2022-01-20T19:51:41Z

Lorentz force

Khanh: /* Applications */

2022-01-20T19:50:48Z

Applications

Khanh: /* Alternative ways to compute */

2022-01-20T19:41:39Z

Alternative ways to compute

Khanh: /* Alternative formulation */

2022-01-20T19:34:45Z

Alternative formulation

Khanh: /* Coordinate notation */

2022-01-20T19:33:12Z

Coordinate notation

Khanh at 19:28, 20 January 2022

2022-01-20T19:28:17Z

Show changes

Khanh: /* Definition */

2022-01-20T19:09:12Z

Definition

Khanh at 19:08, 20 January 2022

2022-01-20T19:08:03Z

Show changes

@@ Line 49: / Line 49: @@
 ===Coordinate notation===
 [[File:3D Vector.svg|300px|thumb|right|Standard basis vectors ('''i''', '''j''', '''k''', also denoted '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub>) and vector components of '''a''' ('''a'''<sub>x</sub>, '''a'''<sub>y</sub>, '''a'''<sub>z</sub>, also denoted '''a'''<sub>1</sub>, '''a'''<sub>2</sub>, '''a'''<sub>3</sub>)]]
-If ('''i''', '''j''','''k''') is a positively oriented orthonormal basis, the basis vectors satisfy the following equalities<ref name=":1" />
+If ('''i''', '''j''','''k''') is a positively oriented orthonormal basis, the basis vectors satisfy the following equalities
 :<math>\begin{alignat}{2}
    \mathbf{\color{blue}{i}}&\times\mathbf{\color{red}{j}} &&= \mathbf{\color{green}{k}}\\

@@ Line 320: / Line 320: @@
 The cross product conveniently describes the infinitesimal generators of rotations in '''R'''<sup>3</sup>. Specifically, if '''n''' is a unit vector in '''R'''<sup>3</sup> and ''R''(''φ'', '''n''') denotes a rotation about the axis through the origin specified by '''n''', with angle φ (measured in radians, counterclockwise when viewed from the tip of '''n'''), then
 :<math>\left.{d\over d\phi} \right|_{\phi=0} R(\phi,\boldsymbol{n}) \boldsymbol{x} = \boldsymbol{n} \times \boldsymbol{x}</math>
-for every vector '''x''' in '''R'''<sup>3</sup>. The cross product with '''n''' therefore describes the infinitesimal generator of the rotations about '''n'''. These infinitesimal generators form the [[Lie algebra]] '''so'''(3) of the [[rotation group SO(3)]], and we obtain the result that the Lie algebra '''R'''<sup>3</sup> with cross product is isomorphic to the Lie algebra '''so'''(3).
+for every vector '''x''' in '''R'''<sup>3</sup>. The cross product with '''n''' therefore describes the infinitesimal generator of the rotations about '''n'''. These infinitesimal generators form the Lie algebra '''so'''(3) of the rotation group SO(3), and we obtain the result that the Lie algebra '''R'''<sup>3</sup> with cross product is isomorphic to the Lie algebra '''so'''(3).
 == Alternative ways to compute ==

@@ Line 560: / Line 560: @@
 ===Lorentz force===
-The cross product is used to describe the [[Lorentz force]] experienced by a moving electric charge {{math|''q<sub>e</sub>''}}:
+The cross product is used to describe the Lorentz force experienced by a moving electric charge {{math|''q<sub>e</sub>''}}:
 : <math>\mathbf{F} = q_e \left( \mathbf{E}+ \mathbf{v} \times \mathbf{B} \right)</math>

@@ Line 522: / Line 522: @@
 === Computational geometry ===
-The cross product appears in the calculation of the distance of two [[Skew lines#Distance|skew lines]] (lines not in the same plane) from each other in three-dimensional space.
+The cross product appears in the calculation of the distance of two skew lines (lines not in the same plane) from each other in three-dimensional space.
-The cross product can be used to calculate the normal for a triangle or polygon, an operation frequently performed in [[computer graphics]]. For example, the winding of a polygon (clockwise or anticlockwise) about a point within the polygon can be calculated by triangulating the polygon (like spoking a wheel) and summing the angles (between the spokes) using the cross product to keep track of the sign of each angle.
+The cross product can be used to calculate the normal for a triangle or polygon, an operation frequently performed in computer graphics. For example, the winding of a polygon (clockwise or anticlockwise) about a point within the polygon can be calculated by triangulating the polygon (like spoking a wheel) and summing the angles (between the spokes) using the cross product to keep track of the sign of each angle.
-In [[computational geometry]] of [[the plane]], the cross product is used to determine the sign of the [[acute angle]] defined by three points <math> p_1=(x_1,y_1), p_2=(x_2,y_2)</math> and <math> p_3=(x_3,y_3)</math>. It corresponds to the direction (upward or downward) of the cross product of the two coplanar [[vector (geometry)|vector]]s defined by the two pairs of points <math>(p_1, p_2)</math> and <math>(p_1, p_3)</math>. The sign of the acute angle is the sign of the expression
+In computational geometry of the plane, the cross product is used to determine the sign of the acute angle defined by three points <math> p_1=(x_1,y_1), p_2=(x_2,y_2)</math> and <math> p_3=(x_3,y_3)</math>. It corresponds to the direction (upward or downward) of the cross product of the two coplanar vectors defined by the two pairs of points <math>(p_1, p_2)</math> and <math>(p_1, p_3)</math>. The sign of the acute angle is the sign of the expression
 :<math> P = (x_2-x_1)(y_3-y_1)-(y_2-y_1)(x_3-x_1),</math>
 which is the signed length of the cross product of the two vectors.
-In the "right-handed" coordinate system,  if the result is 0, the points are [[collinear]]; if it is positive, the three points constitute a positive angle of rotation around <math> p_1</math> from <math> p_2</math> to <math> p_3</math>, otherwise a negative angle. From another point of view, the sign of <math>P</math> tells whether <math> p_3</math> lies to the left or to the right of line <math> p_1, p_2.</math>
+In the "right-handed" coordinate system,  if the result is 0, the points are collinear; if it is positive, the three points constitute a positive angle of rotation around <math> p_1</math> from <math> p_2</math> to <math> p_3</math>, otherwise a negative angle. From another point of view, the sign of <math>P</math> tells whether <math> p_3</math> lies to the left or to the right of line <math> p_1, p_2.</math>
-The cross product is used in calculating the volume of a [[polyhedron]] such as a [[tetrahedron#Volume|tetrahedron]] or [[parallelepiped#Volume|parallelepiped]].
+The cross product is used in calculating the volume of a polyhedron such as a tetrahedron or parallelepiped.
 ===Angular momentum and torque===
-The [[angular momentum]] {{math|'''L'''}} of a particle about a given origin is defined as:
+The angular momentum {{math|'''L'''}} of a particle about a given origin is defined as:
 :  <math>\mathbf{L} = \mathbf{r} \times \mathbf{p},</math>
@@ Line 542: / Line 542: @@
 where {{math|'''r'''}} is the position vector of the particle relative to the origin, {{math|'''p'''}} is the linear momentum of the particle.
-In the same way, the [[Moment (physics)|moment]] {{math|'''M'''}} of a force {{math|'''F'''<sub>B</sub>}} applied at point B around point A is given as:
+In the same way, the moment {{math|'''M'''}} of a force {{math|'''F'''<sub>B</sub>}} applied at point B around point A is given as:
 :  <math> \mathbf{M}_\mathrm{A} = \mathbf{r}_\mathrm{AB} \times \mathbf{F}_\mathrm{B}\,</math>
-In mechanics the ''moment of a force'' is also called ''[[torque]]'' and written as <math>\mathbf{\tau}</math>
+In mechanics the ''moment of a force'' is also called ''torque'' and written as <math>\mathbf{\tau}</math>
-Since position {{nowrap|{{math|'''r'''}},}} linear momentum {{math|'''p'''}} and force {{math|'''F'''}} are all ''true'' vectors, both the angular momentum {{math|'''L'''}} and the moment of a force {{math|'''M'''}} are ''pseudovectors'' or ''axial vectors''.
+Since position {{math|'''r'''}}, linear momentum {{math|'''p'''}} and force {{math|'''F'''}} are all ''true'' vectors, both the angular momentum {{math|'''L'''}} and the moment of a force {{math|'''M'''}} are ''pseudovectors'' or ''axial vectors''.
 ===Rigid body===
-The cross product frequently appears in the description of rigid motions. Two points ''P''  and ''Q'' on a [[rigid body]] can be related by:
+The cross product frequently appears in the description of rigid motions. Two points ''P''  and ''Q'' on a rigid body can be related by:
 : <math>\mathbf{v}_P - \mathbf{v}_Q = \boldsymbol\omega \times \left( \mathbf{r}_P - \mathbf{r}_Q \right)\,</math>
-where <math>\mathbf{r}</math> is the point's position, <math>\mathbf{v}</math> is its velocity and <math>\boldsymbol\omega</math> is the body's [[angular velocity]].
+where <math>\mathbf{r}</math> is the point's position, <math>\mathbf{v}</math> is its velocity and <math>\boldsymbol\omega</math> is the body's angular velocity.
 Since position <math>\mathbf{r}</math> and velocity <math>\mathbf{v}</math> are ''true'' vectors, the angular velocity <math>\boldsymbol\omega</math> is a ''pseudovector'' or ''axial vector''.
 ===Lorentz force===
-{{see also|Lorentz force}}
+The cross product is used to describe the [[Lorentz force]] experienced by a moving electric charge {{math|''q<sub>e</sub>''}}:
 : <math>\mathbf{F} = q_e \left( \mathbf{E}+ \mathbf{v} \times \mathbf{B} \right)</math>
-Since velocity {{nowrap|{{math|'''v'''}},}} force {{math|'''F'''}} and electric field {{math|'''E'''}} are all ''true'' vectors, the magnetic field {{math|'''B'''}} is a ''pseudovector''.
+Since velocity {{math|'''v'''}}, force {{math|'''F'''}} and electric field {{math|'''E'''}} are all ''true'' vectors, the magnetic field {{math|'''B'''}} is a ''pseudovector''.
 === Other ===
-In [[vector calculus]], the cross product is used to define the formula for the [[vector operator]] [[Curl (mathematics)|curl]].
+In vector calculus, the cross product is used to define the formula for the vector operator curl.
-The trick of rewriting a cross product in terms of a matrix multiplication appears frequently in [[epipolar geometry|epipolar]] and multi-view geometry, in particular when deriving matching constraints.
+The trick of rewriting a cross product in terms of a matrix multiplication appears frequently in epipolar and multi-view geometry, in particular when deriving matching constraints.
 == As an external product ==

@@ Line 333: / Line 333: @@
 \end{align}</math>
-where superscript {{math|{{sup|T}}}} refers to the transpose operation, and ['''a''']<sub>×</sub> is defined by:
+where superscript <sup>T</sup> refers to the transpose operation, and ['''a''']<sub>×</sub> is defined by:
 :<math>[\mathbf{a}]_{\times} \stackrel{\rm def}{=} \begin{bmatrix}\,\,0&\!-a_3&\,\,\,a_2\\\,\,\,a_3&0&\!-a_1\\\!-a_2&\,\,a_1&\,\,0\end{bmatrix}.</math>
@@ Line 345: / Line 345: @@
 :<math>[\mathbf{a}]_{\times} = \sum_{i=1}^3\left(\mathbf{a} \times \mathbf{\hat{e}_i}\right)\otimes\mathbf{\hat{e}_i},</math>
-where <math>\otimes</math> is the [[outer product]] operator.
+where <math>\otimes</math> is the outer product operator.
 Also, if '''a''' is itself expressed as a cross product:
@@ Line 355: / Line 355: @@
 :<math>[\mathbf{a}]_{\times} = \mathbf{d}\mathbf{c}^\mathrm{T} - \mathbf{c}\mathbf{d}^\mathrm{T} .</math>
-:{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
-!Proof by substitution
+'''Proof by substitution'''
-|-
-|Evaluation of the cross product gives
+Evaluation of the cross product gives
 :<math> \mathbf{a} = \mathbf{c} \times \mathbf{d} = \begin{pmatrix}
 c_2 d_3 - c_3 d_2 \\
@@ Line 390: / Line 390: @@
 </math>
 Comparison shows that the left hand side equals the right hand side.
-|}
+</blockquote>
 This result can be generalized to higher dimensions using geometric algebra. In particular in any dimension bivectors can be identified with skew-symmetric matrices, so the product between a skew-symmetric matrix and vector is equivalent to the grade-1 part of the product of a bivector and vector. In three dimensions bivectors are dual to vectors so the product is equivalent to the cross product, with the bivector instead of its vector dual. In higher dimensions the product can still be calculated but bivectors have more degrees of freedom and are not equivalent to vectors.
@@ Line 408: / Line 408: @@
 As mentioned above, the Lie algebra '''R'''<sup>3</sup> with cross product is isomorphic to the Lie algebra '''so(3)''', whose elements can be identified with the 3×3 skew-symmetric matrices. The map '''a''' → ['''a''']<sub>×</sub> provides an isomorphism between '''R'''<sup>3</sup> and '''so(3)'''. Under this map, the cross product of 3-vectors corresponds to the commutator of 3x3 skew-symmetric matrices.
-:{| class="toccolours collapsible collapsed" width="70%" style="text-align:left"
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
-!Matrix conversion for cross product with canonical base vectors
+'''Matrix conversion for cross product with canonical base vectors'''
-|-
-|Denoting with <math>\mathbf{e}_i \in \mathbf{R}^{3 \times 1}</math> the <math>i</math>-th canonical base vector, the cross product of a generic vector <math>\mathbf{v} \in \mathbf{R}^{3 \times 1}</math> with <math>\mathbf{e}_i</math> is given by: <math>\mathbf{v} \times \mathbf{e}_i = \mathbf{C}_i \mathbf{v}</math>, where
+Denoting with <math>\mathbf{e}_i \in \mathbf{R}^{3 \times 1}</math> the <math>i</math>-th canonical base vector, the cross product of a generic vector <math>\mathbf{v} \in \mathbf{R}^{3 \times 1}</math> with <math>\mathbf{e}_i</math> is given by: <math>\mathbf{v} \times \mathbf{e}_i = \mathbf{C}_i \mathbf{v}</math>, where
 <math> \mathbf{C}_1 = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{bmatrix}, \quad
@@ Line 424: / Line 424: @@
 * <math>\mathbf{C}_i \mathbf{C}_i^\textrm{T} = \mathbf{P}^{ ^\perp}_{\mathbf{e}_i}</math> (see below);
-The [[projection (linear algebra)#Orthogonal projection|orthogonal projection matrix]] of a vector <math>\mathbf{v} \neq \mathbf{0}</math> is given by <math>\mathbf{P}_{\mathbf{v}} = \mathbf{v}\left(\mathbf{v}^\textrm{T} \mathbf{v}\right)^{-1} \mathbf{v}^T</math>. The projection matrix onto the [[orthogonal complement]] is given by <math>\mathbf{P}^{ ^\perp}_{\mathbf{v}} = \mathbf{I} - \mathbf{P}_{\mathbf{v}}</math>, where <math>\mathbf{I}</math> is the identity matrix. For the special case of <math>\mathbf{v} = \mathbf{e}_i</math>, it can be verified that
+The orthogonal projection matrix of a vector <math>\mathbf{v} \neq \mathbf{0}</math> is given by <math>\mathbf{P}_{\mathbf{v}} = \mathbf{v}\left(\mathbf{v}^\textrm{T} \mathbf{v}\right)^{-1} \mathbf{v}^T</math>. The projection matrix onto the orthogonal complement is given by <math>\mathbf{P}^{ ^\perp}_{\mathbf{v}} = \mathbf{I} - \mathbf{P}_{\mathbf{v}}</math>, where <math>\mathbf{I}</math> is the identity matrix. For the special case of <math>\mathbf{v} = \mathbf{e}_i</math>, it can be verified that
 <math> \mathbf{P}^{^\perp}_{\mathbf{e}_1} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \quad
@@ Line 431: / Line 431: @@
 </math>
-For other properties of orthogonal projection matrices, see [[projection (linear algebra)]].
+For other properties of orthogonal projection matrices, see projection (linear algebra).
-|}
+</blockquote>
 ===Index notation for tensors===

@@ Line 267: / Line 267: @@
 :<math>  \left\| \mathbf{a \times b} \right\|^2 = \left\| \mathbf{a} \right\| ^2 \left\| \mathbf{b}\right \| ^2 \left(1-\cos^2 \theta \right) .</math>
-Invoking the [[Pythagorean trigonometric identity]] one obtains:
+Invoking the Pythagorean trigonometric identity one obtains:
 :<math> \left\| \mathbf{a} \times \mathbf{b} \right\| = \left\| \mathbf{a} \right\| \left\| \mathbf{b} \right\| \left| \sin \theta \right| ,</math>

@@ Line 48: / Line 48: @@
 ===Coordinate notation===
-[[File:3D Vector.svg|300px|thumb|right|[[Standard basis]] vectors ('''i''', '''j''', '''k''', also denoted '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub>) and vector components of '''a''' ('''a'''<sub>x</sub>, '''a'''<sub>y</sub>, '''a'''<sub>z</sub>, also denoted '''a'''<sub>1</sub>, '''a'''<sub>2</sub>, '''a'''<sub>3</sub>)]]
+[[File:3D Vector.svg|300px|thumb|right|Standard basis vectors ('''i''', '''j''', '''k''', also denoted '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub>) and vector components of '''a''' ('''a'''<sub>x</sub>, '''a'''<sub>y</sub>, '''a'''<sub>z</sub>, also denoted '''a'''<sub>1</sub>, '''a'''<sub>2</sub>, '''a'''<sub>3</sub>)]]
 If ('''i''', '''j''','''k''') is a positively oriented orthonormal basis, the basis vectors satisfy the following equalities<ref name=":1" />
 :<math>\begin{alignat}{2}
@@ Line 56: / Line 56: @@
 \end{alignat}</math>
-which imply, by the [[anticommutativity]] of the cross product, that
+which imply, by the anticommutativity of the cross product, that
 :<math>\begin{alignat}{2}
    \mathbf{\color{red}{j}}&\times\mathbf{\color{blue}{i}} &&= -\mathbf{\color{green}{k}}\\

@@ Line 32: / Line 32: @@
 [[File:Cross product.gif|left|thumb|The cross product '''a''' × '''b''' (vertical, in purple) changes as the angle between the vectors '''a''' (blue) and '''b''' (red) changes. The cross product is always orthogonal to both vectors, and has magnitude zero when the vectors are parallel and maximum magnitude ‖'''a'''‖‖'''b'''‖ when they are orthogonal.]]
-By convention, the direction of the vector '''n''' is given by the right-hand rule, where one simply points the forefinger of the right hand in the direction of '''a''' and the middle finger in the direction of '''b'''. Then, the vector '''n''' is coming out of the thumb (see the adjacent picture). Using this rule implies that the cross product is [[Anticommutativity|anti-commutative]]; that is, '''b''' × '''a''' = −('''a''' × '''b'''). By pointing the forefinger toward '''b''' first, and then pointing the middle finger toward '''a''', the thumb will be forced in the opposite direction, reversing the sign of the product vector.
+By convention, the direction of the vector '''n''' is given by the right-hand rule, where one simply points the forefinger of the right hand in the direction of '''a''' and the middle finger in the direction of '''b'''. Then, the vector '''n''' is coming out of the thumb (see the adjacent picture). Using this rule implies that the cross product is anti-commutative; that is, '''b''' × '''a''' = −('''a''' × '''b'''). By pointing the forefinger toward '''b''' first, and then pointing the middle finger toward '''a''', the thumb will be forced in the opposite direction, reversing the sign of the product vector.
 As the cross product operator depends on the orientation of the space (as explicit in the definition above), the cross product of two vectors is not a "true" vector, but a ''pseudovector''.