Cross product

In mathematics, the cross product is a binary operation on vectorss in three dimensions. It is also known as the vector product or outer product. It differs from the dot product in that it results in a vector rather than in a scalar. Its main use lies in the fact that the cross product of two vectors is perpendicular to both of them.

Table of contents

1 Definition
2 Properties
3 Applications
4 Higher dimensions

Definition

The cross product of the two vectors a and b is denoted by a×b (in longhand some mathematicians write a^b to avoid confusion with the letter x); it is defined as:

where θ is the measure of the angle between a and b (between 0 and 180 degrees, or between 0 and π if measured in radian), and n is a unit vector perpendicular to both a and b.

The problem with this definition is that there are two unit vectors perpendicular to both a and b: if n is perpedicular, then so is -n.

Which vector is the correct one depends upon the orientation of the vector space, i.e. on the handedness of the given orthogonal coordinate system i, j, k. In a right-handed system a×b is defined so that a, b and a×b also becomes a right handed system. If i, j, k is left-handed, then a, b and a×b is defined to be left-handed. Because the cross product depends on the choice of coordinate systems, its result is referred to as a pseudovector. Fortunately, in nature cross products tend to come in pairs, so that the "handedness" of the coordinate system is undone by a second cross product.

The cross product can be represented graphically:

Properties

The length of the cross product, | a×b | can be interpreted as the area of the parallelogram having a and b as sides. This means that the triple product gives the volume of the parallelepiped formed by a b and c.

The cross product is anti-symmetric, which means:

a×b = -b×a

The cross product is distributive across addition, meaning that

a×(b + c) = a×b + a×c

It is compatible with scalar multiplication in the following sense:

(ra)×b = a×(rb) = r(a×b).

Two non-zero vectors a and b are parallel if and only if a×b = 0.

The cross product is not associative, but satisfies the Jacobi identity:

a×(b×c) + b×(c×a) + c×(a×b) = 0

and Lagrange's formula:

a×(b×c) = (a·c)b - (a·b)c

The distributivity, linearity and Jacobi identity show that R³ together with vector addition and cross product forms a Lie algebra.

The unit vectors i, j, k from the given orthogonal coordinate system satisfy:

i×j = k, j×k = i, k×i = j

With these rules, the coordinates of the cross product of two vectors can be computed easily, without the need to determine any angles: if a = a₁i + a₂j + a₃k = [a₁, a₂, a₃] and b = b₁i + b₂j + b₃k = [b₁, b₂, b₃] then

a×b = [a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁]

The above component notation can also be written formally as the determinant of a matrix:

The determinant of three vectors can be recovered as

det(a,b,c) = a·(b×c).

The cross product can also be described in terms of quaternions. Notice for instance that the above given cross product relations among i, j and k agree with the multiplicative relations among the quaternions i, j and k. In general, if we represent a vector [a₁, a₂, a₃] as the quaternion a₁ i + a₂ j + a₃ k, we obtain the cross product of two vectors by taking their product as quaternions and deleting the real part of the result (the real part will be the negative of the dot product of the two vectors). More about the connection between quaternion multiplication, vector operations and geometry can be found at quaternions and spatial rotation.

Applications

The cross product occurs in the formula for the vector operator curl. The cross product is also used to describe the Lorenz force experienced by a moving electrical charge in a magnetic field. The definitions of torque and angular momentum also involve the cross product.

Higher dimensions

A cross product for 7-dimensional vectors can be obtained in the same way by using the octonions instead of the quaternions.

This 7-dimensional cross product has the following properties in common with the usual 3-dimensional cross product:

It is bilinear in the sense that x×(ay+bz) = ax×y+bx×z and (ay+bz)×x = ay×x+bz×x
It is anti-commutative: x×y + y×x = 0
It is perpendicular to both x and y: x·(x×y) = y·(x×y) = 0
It satisfies the Jacobi identity: x×(y×z) + y×(z×x) + z×(x×y) = 0
We have ||x×y||² = ||x||²||y||²-(x·y)²