Cross product

In mathematics, the cross product is a binary operation on vectors in vector space. 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.

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 can be defined by

$\mathbf {a} \times \mathbf {b} =\mathbf {\hat {n}} \left|\mathbf {a} \right|\left|\mathbf {b} \right|\sin \theta$

where θ is the measure of the angle between a and b (0° ≤ θ ≤ 180°) on the plane defined by the span of the vectors, 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 perpendicular, 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). The cross product a × b is defined in such a way that (a, b, a × b) becomes right-handed if (i, j, k) is right-handed, or left-handed if (i, j, k) is left-handed.

An easy way to compute the direction of the resultant vector is the "left-hand rule." If the system is right-handed, one simply points the left thumb in the direction of the first operand and the left middle finger in the direction of the second operand's perpendicular component. Then, the resultant vector is coming out of the top of the left hand.

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, with respect to a right-handed coordinate system, as follows:

Properties

Geometric meaning

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.

Algebraic properties

The cross product is anticommutative,

a × b = -b × a,

distributive over addition,

a × (b + c) = a × b + a × c,

and compatible with scalar multiplication so that

(ra) × b = a × (rb) = r(a × b).

It is not associative, but satisfies the Jacobi identity:

a × (b × c) + b × (c × a) + c × (a × b) = 0

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

Further, two non-zero vectors a and b are parallel iff a × b = 0.

Lagrange's formula

This is a well-known and useful formula,

a × (b × c) = b(a · c) − c(a · b),

which is easier to remember as “BAC minus CAB”. This formula is very useful in simplifying vector calculations in physics. It is important to note, however, that it does not hold when involving the del (nabla) operator.

A special case regarding gradients and useful in vector calculus, is

${\begin{matrix}\nabla \times (\nabla \times \mathbf {f} )&=&\nabla (\nabla \cdot \mathbf {f} )-(\nabla \cdot \nabla )\mathbf {f} \\&=&{\mbox{grad }}({\mbox{div }}\mathbf {f} )-{\mbox{laplacian }}\mathbf {f} .\end{matrix}}$

This is a special case of the more general Hodge decomposition $\Delta =d\partial +\partial d$ of the Hodge Laplacian.

Another useful identity of Lagrange is

$|a\times b|^{2}+|a\cdot b|^{2}=|a|^{2}|b|^{2}.$

This is a special case of the multiplicativity $|vw|=|v||w|$ of the norm in the quaternion algebra.

Matrix notation

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

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: Let

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:

$\mathbf {a} \times \mathbf {b} =\det {\begin{bmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\\end{bmatrix}}$

The determinant of three vectors can be recovered as

det (a, b, c) = a · (b × c).

Intuitively, the cross product can be described by Sarrus's scheme where

${\begin{matrix}\mathbf {i} &\mathbf {j} &\mathbf {k} &\mathbf {i} &\mathbf {j} &\mathbf {k} \\a_{1}&a_{2}&a_{3}&a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}&b_{1}&b_{2}&b_{3}\end{matrix}}$

For the first three unit vectors, multiply the elements on the diagonal to the right (e.g. the first diagonal would contain i, a₂, and b₃). For the last three unit vectors, multiply the elements on the diagonal to the left and then negate the product (e.g. the last diagonal would contain k, a₂, and b₁). The cross product would be defined by the sum of these products:

$\mathbf {i} (a_{2}b_{3})+\mathbf {j} (a_{3}b_{1})+\mathbf {k} (a_{1}b_{2})-\mathbf {i} (a_{3}b_{2})-\mathbf {j} (a_{1}b_{3})-\mathbf {k} (a_{2}b_{1})$

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 a₁i + a₂j + a₃k, 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. It is also used to describe the Lorentz force experienced by a moving electrical charge in a magnetic field. The definitions of torque and angular momentum also involve the cross product.

The cross product can also be used to calculate the normal for a triangle or polygon.

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

(ay + bz) × x = ay × x + bz × x.

• It is anticommutative:

x × y + y × x = 0

• It is perpendicular to both x and y:

x · (x × y) = y · (x × y) = 0

• We have

|x × y|² = |x|² |y|² − (x · y)².

Unlike the 3-dimensional cross product, it does not however satisfy the Jacobi identity (equality would hold in 3 dimensions):

x × (y × z) + y × (z × x) + z × (x × y) ≠ 0

In general dimension, there is no direct analogue of the cross product. There is however the wedge product, which has similar properties, except that the wedge product of two vectors is now a 2-vector instead of an ordinary vector. The cross product can be interpreted as the wedge product in three dimensions after using Hodge duality to identify 2-vectors with vectors.

The wedge product and dot product can be combined to form the Clifford product.

See also

• Right-handed rule
• Evaluating cross products