Cross product

From Wikipedia, the free encyclopedia

For the cross product in algebraic topology, see Künneth theorem.

For the crossed product in algebra and functional analysis, see crossed product.

In mathematics, the cross product is a binary operation on vectors in a three-dimensional Euclidean space. It is also known as the vector product or outer product, and is closely related to the exterior 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 orthogonal to both of them.

1 Definition
2 Properties
3 Applications
4 Higher dimensions
5 See also
6 External links

[edit] Definition

The cross product of the two vectors a and b is denoted by a × b (in written form, some mathematicians use a∧b to avoid confusion with the letter x — this should not be confused with the logical "and" operator, $\and$ ). It is defined as the vector which is perpendicular to both a and b with a magnitude equal to the area of the parallelogram they span. The corresponding formula is

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

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, a and b are the scalar magnitudes of vectors a and b respectively, and $\mathbf{\hat{n}}$ is a unit vector perpendicular to both a and b (sometimes the circumflex above the n is omitted). If a and b are parallel or anti-parallel to each other, or one or both of them are null vectors, then the resultant scalar is zero and thus the right hand side of the equation is a null vector (0).

The problem with this definition is that there are two unit vectors perpendicular to both a and b: if $\mathbf{\hat{n}}$ is perpendicular, then so is $-\mathbf{\hat{n}}$ .

Which vector is the "correct" one by convention 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, so i×j=k, j×k= i and k×i=j.

An easy way to compute the direction of the resultant vector is the "right-hand rule" (right hand grip or right hand screw rule may also be used). If the coordinate system is right-handed, one simply points the forefinger of the right hand in the direction of the first operand and the middle finger in the direction of the second operand. Then, the resultant vector is coming out of the thumb.

Because the cross product depends on the choice of coordinate system, its result is referred to as a pseudovector. Fortunately, in nature, measurable quantities involve pairs of cross products, so that the “handedness” of the coordinate system is undone by a second cross product, and the measurement doesn't depend on an arbitrary choice of coordinates.

The cross product can be represented graphically, with respect to a right-handed coordinate system, as shown in the picture below.

[edit] Properties

[edit] Geometric meaning

The magnitude of the cross product can be interpreted as the unsigned area of the parallelogram having a and b as sides:

$| \mathbf{a} \times \mathbf{b}| = | \mathbf{a} | | \mathbf{b}| \sin \theta \,\!$

The triple product gives the signed volume V of the parallelepiped formed by a, b, and c:

$V = \mathbf{a}\cdot(\mathbf{b} \times \mathbf{c}) . \,\!$

[edit] Example 1

Consider two vectors, a = (1,2,3) and b = (4,5,6). The cross product a × b is

a × b = (1,2,3) × (4,5,6) = ((2 × 6 - 3 × 5),(-1 × 6 + 3 × 4), (1 × 5 - 2 × 4)) = (-3,6,-3).

[edit] Example 2

Consider two vectors, a = (3,0,0) and b = (0,2,0). The cross product a × b is

a × b = (3,0,0) × (0,2,0) = ((0 × 0 - 2 × 0), -(3 × 0 - 0 × 0), (3 × 2 - 0 × 0)) = (0,0,6).

This answer has the following interpretations:

The area of the parallelogram (a rectangle in this case) is 2 × 3 = 6.
The second vector lies counterclockwise of the first vector since the answer is positive.
The normal vector to any two vectors in the xy-plane will point in the z direction.

[edit] 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.

It does not obey the cancellation law:

If a × b = a × c and a ≠ 0 then we can write:

(a × b) − (a × c) = 0 and, by the distributive law above:

a × (b − c) = 0

Now, if a is parallel to (b − c), then even if a ≠ 0 it is possible that (b − c) ≠ 0 and therefore that b ≠ c.

However, if both a · b = a · c and a × b = a × c, then we can conclude that b = c. This is because if (b − c) ≠ 0, then it obviously cannot be both parallel and perpendicular to another nonzero vector a.

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.

[edit] 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. Consider the table

$\begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} & \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 & a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 & b_1 & b_2 & b_3 \end{matrix}$

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:

$\mathbf{i}(a_2b_3) + \mathbf{j}(a_3b_1) + \mathbf{k}(a_1b_2) - \mathbf{i}(a_3b_2) - \mathbf{j}(a_1b_3) - \mathbf{k}(a_2b_1).$

Although written here in terms of coordinates, it follows from the geometrical definition above that the cross product is invariant under rotations about the axis defined by $\mathbf{a}\times\mathbf{b}$ , and inverses under swapping $\mathbf{a}$ and $\mathbf{b}$ .

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.

[edit] Conversion to matrix multiplication

A cross product between two vectors in 3d space can be rewritten in terms of pure matrix multiplication as the product of a Skew-symmetric matrix and a vector, as follows:

$\mathbf{a} \times \mathbf{b} = \mathbf{A}_{\times} \mathbf{b} = \begin{bmatrix}0&-a_3&a_2\\a_3&0&-a_1\\-a_2&a_1&0\end{bmatrix}\begin{bmatrix}b_1\\b_2\\b_3\end{bmatrix}$

This notation is often much easier to work with.

[edit] Index notation

The cross product can alternatively be defined in terms of the Levi-Civita tensor $\varepsilon_{ijk}$

$\mathbf{a \times b} = \mathbf{c}\Leftrightarrow\ c_i = \varepsilon_{ijk} a_j b_k$

where the Einstein summation convention is used and the indices $i, j, k$ correspond, as in the previous section, to orthogonal vector components.

[edit] Lagrange's formula

This is an identity involving the cross-product which is very useful. It is written as

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

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

This is a special case of the more general Laplace-de Rham operator $Δ = d δ + δ d$ .

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 $| v w | = | v | | w |$ of the norm in the quaternion algebra.

[edit] 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.

Given a point p and a line through a and b in a plane, all with z coordinate zero, then the z component of (p-a) × (b-a) will be positive or negative, depending on which side of the line p is.

[edit] Higher dimensions

A cross product for 7-dimensional vectors can be obtained in the same way by using the octonions instead of the quaternions. See seven dimensional cross product for the main article. The nonexistence of cross products of two vectors in other dimensions is related to the result that the only normed division algebras are the ones with dimension 1, 2, 4, and 8.

In general dimension, there is no direct analogue of the binary 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.

One can also construct an n-ary analogue of the cross product in Rⁿ⁺¹ given by

$\bigwedge(\mathbf{v}_1,\cdots,\mathbf{v}_n)= \begin{vmatrix} v_1{}^1 &\cdots &v_1{}^{n+1}\\ \vdots &\ddots &\vdots\\ v_n{}^1 & \cdots &v_n{}^{n+1}\\ \mathbf{e}_1 &\cdots &\mathbf{e}_{n+1} \end{vmatrix}.$

This formula is identical in structure to the determinant formula for the normal cross product in R³ except that the row of basis vectors is the last row in the determinant rather than the first. The reason for this is to ensure that the ordered vectors (v₁,...,v_n,Λ(v₁,...,v_n)) have a positive orientation with respect to (e₁,...,e_n+1). If n is even, this modification leaves the value unchanged, so this convention agrees with the normal definition of the binary product. In the case that n is odd, however, the distinction must be kept. This n-ary form enjoys many of the same properties as the vector cross product: it is alternating and linear in its arguments, it is perpendicular to each argument, and its magnitude gives the hypervolume of the region bounded by the arguments. And just like the vector cross product, it can be defined in a coordinate independent way as the Hodge dual of the wedge product of the arguments.

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

In the context of multilinear algebra, it is possible to define a generalized cross product in terms of parity such that the generalized cross product between two vectors of dimension n is a tensor of rank n−2. This is a different concept than what is discussed above.

[edit] See also

Right-hand rule
Evaluating cross products
Lagrange's formula — a common useful cross product identity.
Triple product — a formula using the cross product and dot product
Dot product
Cartesian product — a product of two sets.
Xyzzy in cross products
× (the symbol)

[edit] External links

Multi-dimensional vector product is only possible in seven dimensional space.
Real and Complex Products of Complex Numbers
Vector Product Calculator Online application to calculate the vector product of 3 element vectors
An interactive tutorial created at Syracuse University - (requires java)
Cross-Products and Rotations in Euclidean 2- and 3-Space (PDF)
Multiplication of Vectors

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Categories: Abstract algebra | Linear algebra | Binary operations