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Cross product



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The cross product is a mathematical operation which can be done between two three-dimensional vectors. It is often represented by the symbol [math]\times[/math].[1] After performing the cross product, a new vector is formed. The cross product of two vectors is always perpendicular to both of the vectors which were "crossed".[2][3] This means that cross product is normally only valid in three-dimensional space.

Importance of the cross product

Being a vector operation, the cross product is extremely important in all sorts of sciences (particularly physics), engineering, and mathematics. One important example of the cross product involves torque or moment. Another important application involves the magnetic field.

Visualizing the cross product in three dimensions

Finding the direction of the cross product.

The cross product of [math]\vec{a}[/math] and [math]\vec{b}[/math] is a vector that we shall call [math]\vec c[/math]:

[math] \vec{c}=\vec{a} \times \vec{b}[/math]

The vector [math]\vec a\times\vec b[/math] is perpendicular to both [math]\vec a[/math] and [math]\vec b[/math]. The direction of [math]\vec a\times\vec b[/math] is determined by a variation of the right-hand rule. By holding your right hand as shown in the picture, your thumb points in the direction of [math]\vec c[/math] (the cross product of [math]\vec a[/math] and [math]\vec b[/math]), with the index finger pointing in the direction that [math]\vec a[/math] points, and the middle finger pointing in the direction that [math]\vec b[/math] points. If the angle between the index and middle fingers is greater than 180°, then you need to turn the hand upside down.

How to calculate the cross product in vector notation

Like any mathematical operation, the cross product can be done in a straightforward way.

Two dimensions

Since cross products are usually only defined for three-dimensional vectors, the calculation of cross product in two dimensions treat the vectors as if they are vectors on the xy-plane in three dimension.

More specifically, if
[math]\vec{a} = \langle a_1,a_2 \rangle[/math]
and
[math]\vec{b} = \langle b_1,b_2 \rangle[/math]
then
[math]\vec{a} \times \vec{b} = (a_1b_2-a_2b_1)\hat{k}[/math]

or

[math]\vec{a} \times \vec{b} = \vec{c}[/math]
and
[math]\vec{c} = \langle 0,0,a_1b_2-a_2b_1 \rangle = (a_1b_2-a_2b_1)\hat{k}[/math]

where [math]\hat{k}[/math] is just a symbol indicating that the new vector is pointing up (in the z-direction). If one "crosses" two vectors which are both in the xy-plane, then the product, being perpendicular to both vectors, must point in the z direction. If the value of [math]a_1b_2-a_2b_1[/math] is positive, then it points out of the page; if its value is negative, then it points into the page.

Three dimensions

There are two ways to find the cross product of two 3D vectors: with coordinate notation or with angle.

Coordinate notation

Given vectors [math]\vec a[/math] and [math]\vec b[/math], where
[math]\vec{a} = \langle a_1, a_2, a_3 \rangle[/math]
and
[math]\vec{b} = \langle b_1, b_2, b_3 \rangle[/math]

Then the cross product of [math]\vec a[/math] and [math]\vec b[/math] is:
[math]\vec{a} \times \vec{b} = \langle a_2 b_3 - a_3 b_2, a_3 b_1 - a_1 b_3, a_1 b_2 - a_2 b_1 \rangle[/math].[2]

With angle

Given vectors [math]\vec a[/math] and [math]\vec b[/math], where
[math]\vec{a} = \langle a_1, a_2, a_3 \rangle[/math]
and
[math]\vec{b} = \langle b_1, b_2, b_3 \rangle[/math]

Then the cross product of [math]\vec a[/math] and [math]\vec b[/math] is:
[math]\vec{a} \times \vec{b} = \left\| \mathbf{a} \right\| \left\| \mathbf{b} \right\| \sin (\theta) \ \mathbf{n}[/math],[2]
where [math]\theta[/math] is the angle between [math]\vec a[/math] and [math]\vec b[/math], ‖a‖ and ‖b‖ are the magnitudes of vectors [math]\vec{a}[/math] and [math]\vec{b}[/math], and n is a unit vector perpendicular to the plane containing [math]\vec{a}[/math] and [math]\vec{b}[/math].

Basic properties of the cross product

  • Anti-commutativity: [math]\vec{a} \times \vec{b} = -\vec{b} \times \vec{a}[/math][2]
  • Distributivity over addition: [math]\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}[/math][2]
  • Scalar commutavity: [math]c(\vec{a} \times \vec{b}) = (c\vec{a})\times\vec{b} = \vec{a}\times(c\vec{b})[/math]

Related pages

References

  1. "Comprehensive List of Algebra Symbols" (in en-US). 2020-03-25. https://mathvault.ca/hub/higher-math/math-symbols/algebra-symbols/. 
  2. 2.0 2.1 2.2 2.3 2.4 Weisstein, Eric W.. "Cross Product" (in en). https://mathworld.wolfram.com/CrossProduct.html. 
  3. "Cross Product". https://www.mathsisfun.com/algebra/vectors-cross-product.html.