What is the Definition and Properties of a Cross Product in Vector Algebra?

[Greg Bernhardt]

Definition/Summary

The cross product of two vectors $\mathbf{A}$ and $\mathbf{B}$ is a third vector (strictly, a pseudovector or axial vector) $\mathbf{A}\times\mathbf{B}$ 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: $\mathbf{A}\times\mathbf{B}\,=\,-\,\mathbf{B}\times\mathbf{A}$

If the vectors $\mathbf{A}$ and $\mathbf{B}$ are considered as 1-forms, then the wedge product $\mathbf{A}\wedge\mathbf{B}$ is a 2-form (a directed area), and its dual $\ast(\mathbf{A}\wedge\mathbf{B})$ is a dual 1-form, corresponding to the pseudovector $\mathbf{A}\times\mathbf{B}$

Equations

The magnitude of $\mathbf{A}\times\mathbf{B}$ is the area of the paralleogram with sides $\mathbf{A}$ and $\mathbf{B}$:

$$|\mathbf{A}\times\mathbf{B}| = AB \sin \theta$$

In terms of Cartesian components:

$$(\mathbf{A}\times\mathbf{B})_x = A_y B_z - A_z B_y$$

$$(\mathbf{A}\times\mathbf{B})_y = A_z B_x - A_x B_z$$

$$(\mathbf{A}\times\mathbf{B})_z = A_x B_y - A_y B_x$$

The triple product (scalar product) $\mathbf{A}\cdot(\mathbf{B}\times\mathbf{C})$ is the volume of the parallepiped with sides $\mathbf{A}$ $\mathbf{B}$ and $\mathbf{C}$, and therefore:

$$\mathbf{A}\cdot(\mathbf{B}\times\mathbf{C})\,=\,\mathbf{B}\cdot(\mathbf{C}\times\mathbf{A})\,=\,\mathbf{C}\cdot(\mathbf{A}\times\mathbf{B})$$

The repeated cross product $\mathbf{A}\times(\mathbf{B}\times\mathbf{C})$ is a vector perpendicular to $\mathbf{A}$, and is a linear combination of $\mathbf{B}$ and $\mathbf{C}$:

$$\mathbf{A}\times(\mathbf{B}\times\mathbf{C})\,=\,(\mathbf{A}\cdot\mathbf{C})\mathbf{B}\,-\,(\mathbf{A}\cdot\mathbf{B})\mathbf{C}$$

Extended explanation

For two vectors $\mathbf{a} = \left(a_x,a_y,a_z\right)\;\;\mathbf{b}=\left(b_x,b_y,b_z\right)$ in $\mathbb{R}^3$, the cross poduct can be written as the determinant of a 3x3 matrix:

$$\mathbf{a}\times\mathbf{b} = \left|\begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z\end{array}\right|$$

Where $\mathbf{i}, \mathbf{j}, \mathbf{k}$ is a right-handed orthonormal basis.

Polar vectors and pseudovectors:

A polar (ordinary) vector is reversed under an inversion of the coordinate axes: $\mathbf{a}\ \mapsto\ -\mathbf{a}$

However, the cross product of two polar vectors is not reversed: $\mathbf{a}\ \times\ \mathbf{b}\ \mapsto\ (-\mathbf{a})\ \times\ (-\mathbf{b})\ =\ \mathbf{a}\ \times\ \mathbf{b}$

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 pseudovector (axial vector).

Directed area:

A directed area is a flat surface together with a magnitude (its area), and a sign ($\pm$) indicating a direction of rotation within the surface (alternatively, indicating a preferred normal direction).

A directed area is an elementary 2-form (a wedge product of two 1-forms).

A general 2-form is a sum of directed areas.

Its dual (in three-dimensional space) is a 1-form in the dual space, corresponding to a pseudovector normal to the surface and with magnitude equal to its area.

Triple product 3-form pseudoscalars and directed volume:

The wedge product of three 1-forms is a 3-form.

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 0-form, which is 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 psuedoscalar.

The triple product $\mathbf{a}\cdot(\mathbf{b}\,\times\,\mathbf{c})$ of three vectors is the dot product of a vector and a pseudovector, and is the pseudoscalar equal to the wedge product $\mathbf{a}\,\wedge\,\mathbf{b}\,\wedge\,\mathbf{c}$ of the three vectors, in the same order. It is the directed volume of the parallepiped whose sides are those three vectors ("directed" because it includes a sign ($\pm$) 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!

 

[fresh_42]

An interesting read is about the geometry of it:
https://arxiv.org/pdf/1205.5935.pdf
and the connections: cross product - Lie algebra - determinant - volume of parallelepiped

 

[Kashmir]

I've a doubt regarding the pseudo vector part and I'll be thankful if you could clarify.

Definition/Summary

The cross product of two vectors $\mathbf{A}$ and $\mathbf{B}$ is a third vector (strictly, a pseudovector or axial vector) $\mathbf{A}\times\mathbf{B}$ 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: $\mathbf{A}\times\mathbf{B}\,=\,-\,\mathbf{B}\times\mathbf{A}$

If the vectors $\mathbf{A}$ and $\mathbf{B}$ are considered as 1-forms, then the wedge product $\mathbf{A}\wedge\mathbf{B}$ is a 2-form (a directed area), and its dual $\ast(\mathbf{A}\wedge\mathbf{B})$ is a dual 1-form, corresponding to the pseudovector $\mathbf{A}\times\mathbf{B}$

Equations

The magnitude of $\mathbf{A}\times\mathbf{B}$ is the area of the paralleogram with sides $\mathbf{A}$ and $\mathbf{B}$:

$$|\mathbf{A}\times\mathbf{B}| = AB \sin \theta$$

In terms of Cartesian components:

$$(\mathbf{A}\times\mathbf{B})_x = A_y B_z - A_z B_y$$

$$(\mathbf{A}\times\mathbf{B})_y = A_z B_x - A_x B_z$$

$$(\mathbf{A}\times\mathbf{B})_z = A_x B_y - A_y B_x$$

The triple product (scalar product) $\mathbf{A}\cdot(\mathbf{B}\times\mathbf{C})$ is the volume of the parallepiped with sides $\mathbf{A}$ $\mathbf{B}$ and $\mathbf{C}$, and therefore:

$$\mathbf{A}\cdot(\mathbf{B}\times\mathbf{C})\,=\,\mathbf{B}\cdot(\mathbf{C}\times\mathbf{A})\,=\,\mathbf{C}\cdot(\mathbf{A}\times\mathbf{B})$$

The repeated cross product $\mathbf{A}\times(\mathbf{B}\times\mathbf{C})$ is a vector perpendicular to $\mathbf{A}$, and is a linear combination of $\mathbf{B}$ and $\mathbf{C}$:

$$\mathbf{A}\times(\mathbf{B}\times\mathbf{C})\,=\,(\mathbf{A}\cdot\mathbf{C})\mathbf{B}\,-\,(\mathbf{A}\cdot\mathbf{B})\mathbf{C}$$

Extended explanation

For two vectors $\mathbf{a} = \left(a_x,a_y,a_z\right)\;\;\mathbf{b}=\left(b_x,b_y,b_z\right)$ in $\mathbb{R}^3$, the cross poduct can be written as the determinant of a 3x3 matrix:

$$\mathbf{a}\times\mathbf{b} = \left|\begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z\end{array}\right|$$

Where $\mathbf{i}, \mathbf{j}, \mathbf{k}$ is a right-handed orthonormal basis.

Polar vectors and pseudovectors:

A polar (ordinary) vector is reversed under an inversion of the coordinate axes: $\mathbf{a}\ \mapsto\ -\mathbf{a}$

However, the cross product of two polar vectors is not reversed: $\mathbf{a}\ \times\ \mathbf{b}\ \mapsto\ (-\mathbf{a})\ \times\ (-\mathbf{b})\ =\ \mathbf{a}\ \times\ \mathbf{b}$

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 pseudovector (axial vector).

Directed area:

A directed area is a flat surface together with a magnitude (its area), and a sign ($\pm$) indicating a direction of rotation within the surface (alternatively, indicating a preferred normal direction).

A directed area is an elementary 2-form (a wedge product of two 1-forms).

A general 2-form is a sum of directed areas.

Its dual (in three-dimensional space) is a 1-form in the dual space, corresponding to a pseudovector normal to the surface and with magnitude equal to its area.

Triple product 3-form pseudoscalars and directed volume:

The wedge product of three 1-forms is a 3-form.

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 0-form, which is 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 psuedoscalar.

The triple product $\mathbf{a}\cdot(\mathbf{b}\,\times\,\mathbf{c})$ of three vectors is the dot product of a vector and a pseudovector, and is the pseudoscalar equal to the wedge product $\mathbf{a}\,\wedge\,\mathbf{b}\,\wedge\,\mathbf{c}$ of the three vectors, in the same order. It is the directed volume of the parallepiped whose sides are those three vectors ("directed" because it includes a sign ($\pm$) 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!

In a right hand system suppose I've two vectors i, j then their cross product is $\mathbf{i}\ \times\ \mathbf{j}\ =\ \mathbf{k}$

When we invert our coordinate system and then try to calculate the cross product of those two vectors we will have $\mathbf{-i}\ \times\ \mathbf{-j}\ =\ \mathbf{k}$

That means cross product is a polar vector? But that's wrong.

Do we use a left handed system to compute cross product when we invert our coordinate system? Can you please help me, I'm not able to understand it.
Thank you.

 

[romsofia]

I've a doubt regarding the pseudo vector part and I'll be thankful if you could clarify.

In a right hand system suppose I've two vectors i, j then their cross product is $\mathbf{i}\ \times\ \mathbf{j}\ =\ \mathbf{k}$

When we invert our coordinate system and then try to calculate the cross product of those two vectors we will have $\mathbf{-i}\ \times\ \mathbf{-j}\ =\ \mathbf{k}$

That means cross product is a polar vector? But that's wrong.

Do we use a left handed system to compute cross product when we invert our coordinate system? Can you please help me, I'm not able to understand it.
Thank you.

You got a definition backwards, it is said that under an inversion (also known as a parity transformation), that $\vec{x} \rightarrow -\vec{x}$ and as you just said, the cross product remains the same. So, it can't be a polar vector (aka "real vector"), so historically they called it a pseudovector, I believe.