In
mathematics, the
cross
product is a
binary
operation on two
vector in a
three-dimensional
Euclidean space
that results in another vector which is
perpendicular to the plane containing the two
input vectors. The
algebra
defined by the cross product is neither
commutative nor
associative. It contrasts with the
dot product which produces a
scalar result. In many engineering and
physics problems, it is desirable to be able to construct a
perpendicular vector from two existing vectors, and the cross
product provides a means for doing so. The cross product is also
useful as a measure of "perpendicularness"—the magnitude of the
cross product of two vectors is equal to the product of their
magnitudes if they are perpendicular and scales down to zero when
they are parallel. The cross product is also known as the
vector product, or
Gibbs vector product.
The cross product is only defined in three or
seven dimensions. Like the
dot product, it depends on the
metric
of Euclidean space. Unlike the
dot
product, it also depends on the choice of
orientation or "handedness".
Certain features of the cross product can be generalized to other
situations. For arbitrary choices of orientation, the cross product
must be regarded not as a vector, but as a
pseudovector. For arbitrary choices of metric,
and in arbitrary dimensions, the cross product can be generalized
by the
exterior product of vectors,
defining a
two-form instead of a vector.
Definition
The cross product of two vectors
a and
b is denoted by In
physics,
sometimes the notation
a∧
b is
used, though this is avoided in mathematics to avoid confusion with
the
exterior product.
In a three-dimensional
Euclidean
space, with a
right-handed
coordinate system,
a ×
b is
defined as a vector
c that is
perpendicular to both
a and
b, with a direction given by the
right-hand rule and a magnitude equal to the
area of the
parallelogram that the
vectors span.
The cross product is defined by the formula
- \mathbf{a} \times \mathbf{b} = a b \sin \theta \
\mathbf{\hat{n}}
where
θ is the measure of the smaller
angle between
a and
b (0° ≤
θ ≤ 180°),
a and
b are the
magnitudes of
vectors
a and
b, and
\scriptstyle\mathbf{\hat{n}} is a
unit
vector perpendicular to the plane
containing
a and
b in the
direction given by the right-hand rule as illustrated. If the
vectors
a and
b are collinear
(i.e., the angle
θ between them is either 0° or 180°), by
the above formula, the cross product of
a and
b is the zero vector
0.
The direction of the vector \scriptstyle\mathbf{\hat{n}} is given
by the right-hand rule, where one simply points the forefinger of
the right hand in the direction of
a and the
middle finger in the direction of
b. Then, the
vector \scriptstyle\mathbf{\hat{n}} is coming out of the thumb (see
the picture on the right). Using this rule implies that the
cross-product is
anti-commutative,
i.e.,
b ×
a =
-(
a ×
b). By pointing the
forefinger toward
b first, and then pointing the
middle finger toward
a, the thumb will be forced
in the opposite direction, reversing the sign of the product
vector.
Using the cross product requires the handedness of the coordinate
system to be taken into account (as explicit in the definition
above). If a
left-handed
coordinate system is used, the direction of the vector
\scriptstyle\mathbf{\hat{n}} is given by the left-hand rule and
points in the opposite direction.
This, however, creates a problem because transforming from one
arbitrary reference system to another (
e.g., a mirror
image transformation from a right-handed to a left-handed
coordinate system), should not change the direction of
\scriptstyle\mathbf{\hat{n}}. The problem is clarified by realizing
that the cross-product of two vectors is not a (true) vector, but
rather a
pseudovector. See
cross
product and handedness for more detail.
Computing the cross product
Coordinate 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
Together with the skew-symmetry and bilinearity of the cross
product, these three identities are sufficient to determine the
cross product of any two vectors. In particular, the following
identities are also seen to hold
- j × i = −k
k ×
j = −i
i × k =
−j
- i × i = j ×
j = k × k =
0.
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 = a_{1}i +
a_{2}j +
a_{3}k = (a_{1},
a_{2}, a_{3})
and
- b = b_{1}i +
b_{2}j +
b_{3}k = (b_{1},
b_{2}, b_{3}).
The cross product can be calculated by
distributive cross-multiplication:
- a × b =
(a_{1}i +
a_{2}j +
a_{3}k) ×
(b_{1}i +
b_{2}j +
b_{3}k)
- a × b =
a_{1}i ×
(b_{1}i +
b_{2}j +
b_{3}k) +
a_{2}j ×
(b_{1}i +
b_{2}j +
b_{3}k) +
a_{3}k ×
(b_{1}i +
b_{2}j +
b_{3}k)
- a × b =
(a_{1}i ×
b_{1}i) +
(a_{1}i ×
b_{2}j) +
(a_{1}i ×
b_{3}k) +
(a_{2}j ×
b_{1}i) +
(a_{2}j ×
b_{2}j) +
(a_{2}j ×
b_{3}k) +
(a_{3}k ×
b_{1}i) +
(a_{3}k ×
b_{2}j) +
(a_{3}k ×
b_{3}k).
Since
scalar multiplication is
commutative with cross multiplication,
the right hand side can be regrouped as
- a × b =
a_{1}b_{1}(i ×
i) +
a_{1}b_{2}(i ×
j) +
a_{1}b_{3}(i ×
k) +
a_{2}b_{1}(j ×
i) +
a_{2}b_{2}(j ×
j) +
a_{2}b_{3}(j ×
k) +
a_{3}b_{1}(k ×
i) +
a_{3}b_{2}(k ×
j) +
a_{3}b_{3}(k ×
k).
This equation is the sum of nine simple cross products. After all
the multiplication is carried out using the basic cross product
relationships between
i,
j, and
k defined above,
- a × b =
a_{1}b_{1}(0) +
a_{1}b_{2}(k) +
a_{1}b_{3}(−j) +
a_{2}b_{1}(−k) +
a_{2}b_{2}(0) +
a_{2}b_{3}(i) +
a_{3}b_{1}(j) +
a_{3}b_{2}(−i) +
a_{3}b_{3}(0).
This equation can be
factored to form
- a × b =
(a_{2}b_{3} −
a_{3}b_{2}) i +
(a_{3}b_{1} −
a_{1}b_{3}) j +
(a_{1}b_{2} −
a_{2}b_{1}) k =
(a_{2}b_{3} −
a_{3}b_{2},
a_{3}b_{1} −
a_{1}b_{3},
a_{1}b_{2} −
a_{2}b_{1}).
Matrix notation
The definition of the cross product can also be represented by 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}.
This determinant can be computed using
Sarrus' rule. 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} From the first three
elements on the first row draw three diagonals sloping downward to
the right (for example, the first diagonal would contain
i,
a_{2}, and
b_{3}), and from the last three elements on the
first row draw three diagonals sloping downward to the left (for
example, the first diagonal would contain
i,
a_{3}, and
b_{2}). Then multiply
the elements on each of these six diagonals, and negate the last
three products. 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.
Properties
Geometric meaning
Figure 1: The area of a parallelogram as a cross product
Figure 2: The volume of a
parallelepiped using dot and cross-products; dashed lines show the
projections of
c onto
a × b and
of
a onto
b × c, a first step in
finding dot-products.
The magnitude of the cross product can be interpreted as the
positive
area of the
parallelogram having
a and
b as sides (see Figure 1):
- A = | \mathbf{a} \times \mathbf{b}| = | \mathbf{a} | |
\mathbf{b}| \sin \theta. \,\!
Indeed, one can also compute the volume
V of a
parallelepiped having
a,
b and
c as sides by using a
combination of a cross product and a dot product, called
scalar triple product (see Figure 2):
- V = |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|.
Figure 2 demonstrates that this volume can be found in two ways,
showing geometrically that the identity holds that a "dot" and a
"cross" can be interchanged without changing the result. That
is:
- V =\mathbf{a \times b \cdot c} = \mathbf{a \cdot b \times c} \
.
Because the magnitude of the cross product goes by the sine of the
angle between its arguments, the cross product can be thought of as
a measure of "perpendicularness" in the same way that the
dot product is a measure of "parallelness".
Given two
unit vectors, their cross
product has a magnitude of 1 if the two are perpendicular and a
magnitude of zero if the two are parallel.
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
- (r a) × b =
a × (r b) =
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:
- (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 it
can be concluded that
b =
c. Indeed,
- a · (b - c)
= 0, and
- a × (b - c)
= 0
so that
b -
c is
both
parallel and perpendicular to the non-zero vector
a.
This is only possible if
b -
c =
0.
The distributivity, linearity and Jacobi identity show that
R^{3} together with vector addition and
cross product forms a
Lie algebra. In
fact, the Lie algebra is that of the real
orthogonal group in 3 dimensions,
SO.
Further, two non-zero vectors
a and
b are parallel
if and
only if a ×
b =
0.
It follows from the geometrical definition above that the cross
product is invariant under
rotations about
the axis defined by
a×
b.
There is also this property relating cross products and the triple
product:
- (a × b) × (a
× c) = (a · (b ×
c)) a.
The cross product obeys this identity under
matrix transformations:
- (M\mathbf{a}) \times (M\mathbf{b}) = (\det M) M^{-T}(\mathbf{a}
\times \mathbf{b}) \,\!
where \scriptstyle M is a 3 by 3
matrix and \scriptstyle M^{-T} is the
transpose of the
inverse
The cross product of two vectors in 3-D always lies in the
null space of the matrix with the vectors as
rows. In other words
- \mathbf{a} \times \mathbf{b} \in
NS\left(\begin{bmatrix}\mathbf{a} \\
\mathbf{b}\end{bmatrix}\right)
Triple product expansion
The triple product expansion, also known as
Lagrange's
formula, is a formula relating the cross product of three
vectors (called the
vector triple product) with
the dot product:
- a × (b × c)
= b(a · c) −
c(a · b).
The
mnemonic "BAC minus CAB" is used to
remember the order of the vectors in the right hand member. This
formula is used in
physics to simplify
vector calculations. A special case, regarding
gradients and useful in
vector calculus, is given below.
- \begin{align}
\nabla \times (\nabla \times \mathbf{f})& {}= \nabla (\nabla
\cdot \mathbf{f} ) - (\nabla \cdot \nabla) \mathbf{f} \\& {}=
\mbox{grad }(\mbox{div } \mathbf{f} ) - \Delta
\mathbf{f}.\end{align} This is a special case of the more general
Laplace-de
Rham operator \scriptstyle\Delta = d \delta + \delta d.
The following identity also relates the cross product and the dot
product:
- |\mathbf{a} \times \mathbf{b}|^2 + |\mathbf{a} \cdot
\mathbf{b}|^2 = |\mathbf{a}|^2 |\mathbf{b}|^2.
This is a special case of the multiplicativity \scriptstyle
|\mathbf{vw}| = |\mathbf{v}| |\mathbf{w}| of the norm in the
quaternion algebra, and a restriction to
\scriptstyle\mathbb{R}^3 of
Lagrange's identity.
Alternative ways to compute the cross product
Quaternions
The cross product can also be described in terms of
quaternions, and this is why the letters
i,
j,
k are a
convention for the standard basis on \scriptstyle\mathbf{R}^3: it
is thought of as the imaginary quaternions.
For instance, 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 a vector
[
a_{1},
a_{2},
a_{3}] is represented as the quaternion
a_{1}i +
a_{2}j
+
a_{3}k, the cross product of two
vectors can be obtained 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.
Alternatively and more straightforwardly, using the above
identification of the 'purely imaginary' quaternions with
\scriptstyle\mathbf{R}^3, the cross product may be thought of as
half of the
commutator of two
quaternions.
Conversion to matrix multiplication
A cross product between two vectors (which can only be defined in
three-dimensional 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}
- \mathbf{a} \times \mathbf{b} = [\mathbf{b}]^\top_{\times}
\mathbf{a} = \begin{bmatrix}\,0&\,\,b_3&\!-b_2\\
-b_3&0&\,\,b_1\\\,\,b_2&\!-b_1&\,0\end{bmatrix}\begin{bmatrix}a_1\\a_2\\a_3\end{bmatrix}
where
- [\mathbf{a}]_{\times} \stackrel{\rm def}{=}
\begin{bmatrix}\,\,0&\!-a_3&\,\,\,a_2\\\,\,\,a_3&0&\!-a_1\\\!-a_2&\,\,a_1&\,\,0\end{bmatrix}.
Also, if \scriptstyle\mathbf{a} is itself a cross product:
- \mathbf{a} = \mathbf{c} \times \mathbf{d}
then
- [\mathbf{a}]_{\times} = (\mathbf{c}\mathbf{d}^\top)^\top -
\mathbf{c}\mathbf{d}^\top.
This notation provides another way of generalizing cross product to
the higher dimensions by substituting
pseudovectors (such as
angular velocity or
magnetic field) with such skew-symmetric
matrices. It is clear that such physical quantities will have
n(n-1)/2 independent components in n dimensions, which coincides
with number of dimensions for three-dimensional space, and this is
why vectors can be used (and most often are used) to represent such
quantities.
This notation is also often much easier to work with, for example,
in
epipolar geometry.
From the general properties of the cross product follows
immediately that
- [\mathbf{a}]_{\times} \, \mathbf{a} = \mathbf{0} and
\mathbf{a}^\top \, [\mathbf{a}]_{\times} = \mathbf{0}
and from fact that \scriptstyle[\mathbf{a}]_{\times} is
skew-symmetric it follows that
- \mathbf{b}^\top \, [\mathbf{a}]_{\times} \, \mathbf{b} =
0.
The above-mentioned triple product expansion (bac-cab rule) can be
easily proven using this notation.
The above definition of \scriptstyle[\mathbf{a}]_{\times} means
that there is a one-to-one mapping between the set of 3×3
skew-symmetric matrices, also known as the
Lie algebra of
SO, and the operation of taking the cross
product with some vector \scriptstyle\mathbf{a} .
Index notation
The cross product can alternatively be defined in terms of the
Levi-Civita symbol,
\scriptstyle\varepsilon_{ijk}\mathbf{a \times b} =
\mathbf{c}\Leftrightarrow\ c_i = \sum_{j=1}^3 \sum_{k=1}^3
\varepsilon_{ijk} a_j b_kwhere the
indices \scriptstyle i,j,k correspond,
as in the previous section, to orthogonal vector components. This
characterization of the cross product is often expressed more
compactly using the
Einstein summation convention
as\mathbf{a \times b} = \mathbf{c}\Leftrightarrow\ c_i =
\varepsilon_{ijk} a_j b_kin which repeated indices are summed from
1 to 3. Note that this representation is another form of the
skew-symmetric representation of the cross product:
- \varepsilon_{ijk} a_j = [\mathbf{a}]_\times.
In
classical mechanics:
representing the cross-product with the Levi-Civita symbol can
cause mechanical-symmetries to be obvious when physical-systems are
isotropic in space. (Quick example:
consider a particle in a Hooke's Law potential in three-space, free
to oscillate in three dimensions; none of these dimensions are
"special" in any sense, so symmetries lie in the
cross-product-represented angular-momentum which are made clear by
the abovementioned Levi-Civita representation).
Mnemonic
The word
xyzzy can be used to remember the
definition of the cross product.
If
- \mathbf{a} = \mathbf{b} \times \mathbf{c}
where:
\mathbf{a} = \begin{bmatrix}a_x\\a_y\\a_z\end{bmatrix},\mathbf{b} =
\begin{bmatrix}b_x\\b_y\\b_z\end{bmatrix},\mathbf{c} =
\begin{bmatrix}c_x\\c_y\\c_z\end{bmatrix}
then:
- a_x = b_y c_z - b_z c_y \,
- a_y = b_z c_x - b_x c_z \,
- a_z = b_x c_y - b_y c_x. \,
The second and third equations can be obtained from the first by
simply vertically rotating the subscripts,
x →
y
→
z →
x. The problem, of course, is how to
remember the first equation, and two options are available for this
purpose: either to remember the relevant two diagonals of Sarrus's
scheme (those containing
i), or to
remember the
xyzzy sequence.
Since the first diagonal in Sarrus's scheme is just the
main diagonal of the
above-mentioned \scriptstyle 3
\times 3 matrix, the first three letters of the word
xyzzy can be very easily remembered.
Applications
Computational geometry
The cross product can be used to calculate the normal for a
triangle or polygon, an operation frequently performed in
computer graphics.
In
computational geometry of
the plane, the cross product is used to
determine the sign of the
acute angle
defined by three points \scriptstyle p_1=(x_1,y_1), \scriptstyle
p_2=(x_2,y_2) and \scriptstyle p_3=(x_3,y_3). It corresponds to the
direction of the cross product of the two coplanar
vector defined by the pairs of points
\scriptstyle p_1, p_2 and \scriptstyle p_1, p_3, i.e., by the sign
of the expression \scriptstyle P =
(x_2-x_1)(y_3-y_1)-(y_2-y_1)(x_3-x_1). In the "right-handed"
coordinate system, if the result is 0, the points are collinear; if
it is positive, the three points constitute a negative angle of
rotation around \scriptstyle p_2 from \scriptstyle p_1 to
\scriptstyle p_3, otherwise a positive angle. From another point of
view, the sign of \scriptstyle P tells whether \scriptstyle p_3
lies to the left or to the right of line \scriptstyle p_1,
p_2.
Mechanics
Moment of a force
\scriptstyle\mathbf{F_B} applied at point B around point A is given
as:
- : \mathbf{M_A} = \mathbf{r_{AB}} \times \mathbf{F_B} \,
Other
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 trick of rewriting a cross product in terms of a matrix
multiplication appears frequently in epipolar and multi-view
geometry, in particular when deriving matching constraints.
Cross product as an exterior product
The cross product in relation to the exterior product.
In red are the unit normal vector, and the "parallel" unit
bivector.
The cross product can be viewed in terms of the
exterior product. This view allows for a
natural geometric interpretation of the cross product. In
exterior calculus the exterior product (or
wedge product) of two vectors is a
bivector. A bivector is an oriented plane element,
in much the same way that a vector is an oriented line element.
Given two vectors
a and
b, one can view the
bivector
a∧
b as the oriented parallelogram
spanned by
a and
b. The cross product is then
obtained by taking the
Hodge dual of the
bivector
a∧
b, identifying
2-vectors with vectors:
- a \times b = * (a \wedge b) \,.
This can be thought of as the oriented multi-dimensional element
"perpendicular" to the bivector. Only in three dimensions is the
result an oriented line element – a vector – whereas, for example,
in 4 dimensions the Hodge dual of a bivector is two-dimensional –
another oriented plane element. So, in three dimensions only is the
cross product of
a and
b the vector dual to the
bivector
a∧
b: it is perpendicular to the
bivector, with orientation dependent on the coordinate system's
handedness, and has the same magnitude relative to the unit normal
vector as
a∧
b has relative to the unit bivector;
precisely the properties described above.
Cross product and handedness
When measurable quantities involve cross products, the
handedness of the coordinate systems used cannot be
arbitrary. However, when physics laws are written as equations, it
should be possible to make an arbitrary choice of the coordinate
system (including handedness). To avoid problems, one should be
careful to never write down an equation where the two sides do not
behave equally under all transformations that need to be
considered. For example, if one side of the equation is a cross
product of two vectors, one must take into account that when the
handedness of the coordinate system is
not fixed a priori,
the result is not a (true) vector but a
pseudovector. Therefore, for consistency, the
other side
must also be a pseudovector.
More generally, the result of a cross product may be either a
vector or a pseudovector, depending on the type of its operands
(vectors or pseudovectors). Namely, vectors and pseudovectors are
interrelated in the following ways under application of the cross
product:
- * vector × vector = pseudovector
- * vector × pseudovector = vector
- * pseudovector × pseudovector = pseudovector
Because the cross product may also be a (true) vector, it may not
change direction with a mirror image transformation. This happens,
according to the above relationships, if one of the operands is a
(true) vector and the other one is a pseudovector (
e.g.,
the cross product of two vectors). For instance, a
vector triple product involving three
(true) vectors is a (true) vector.
A handedness-free approach is possible using
exterior algebra.
Generalizations
There are several ways to generalize the cross product to the
higher dimensions.
Lie algebra
The cross product can be seen as one of the simplest Lie
products,and is thus generalized by
Lie
algebras, which are axiomatized as binary products satisfying
the axioms of multilinearity, skew-symmetry, and the Jacobi
identity. Many Lie algebras exist, and their study is a major field
of mathematics, called
Lie theory.
For example, the
Heisenberg
algebra gives another Lie algebra structure on
\scriptstyle\mathbf{R}^3. In the basis \scriptstyle\{x,y,z\}, the
product is \scriptstyle [x,y]=z, [x,z]=[y,z]=0.
Using octonions
A cross product for 7-dimensional vectors can be obtained in the
same way by using the
octonions instead of
the quaternions. The nonexistence of such 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.
Wedge product
In general dimension, there is no direct analogue of the binary
cross product. There is however the
wedge
product, whichhas similar properties, except that the wedge
product of two vectors is now a
2-vector
instead of an ordinary vector. As mentioned above, 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.
Multilinear algebra
In the context of
multilinear
algebra, the cross product can be seen as the (1,2)-tensor (a
mixed tensor) obtained from the
3-dimensional
volume form,By a volume
form one means a function that takes in
n vectors and
gives out a scalar, the volume of the
parallelotope defined by the vectors:
\scriptstyle V\times \cdots \times V \to \mathbf{R}. This is an
n-ary multilinear skew-symmetric form. In the presence of
a basis, such as on \scriptstyle\mathbf{R}^n, this is given by the
determinant, but in an abstract vector
space, this is added structure. In terms of
G-structures, a volume form is an
\scriptstyle SL-structure. a
(0,3)-tensor, by
raising an
index.
In detail, the 3-dimensional volume form defines a product
\scriptstyle V \times V \times V \to \mathbf{R}, by taking the
determinant of the matrix given by these 3 vectors.By
duality, this is equivalent to a
function \scriptstyle V \times V \to V^*, (fixing any two inputs
gives a function \scriptstyle V \to \mathbf{R} by evaluating on the
third input) and in the presence of an
inner product (such as the
dot product; more generally, a non-degenerate
bilinear form), we have an isomorphism \scriptstyle V \to V^*, and
thus this yields a map \scriptstyle V \times V \to V, which is the
cross product: a (0,3)-tensor (3 vector inputs, scalar output) has
been transformed into a (1,2)-tensor (2 vector inputs, 1 vector
output) by "raising an index".
Translating the above algebra into geometry, the function "volume
of the parallelepiped defined by \scriptstyle (a,b,-)" (where the
first two vectors are fixed and the last is an input), which
defines a function \scriptstyle V \to \mathbf{R}, can be
represented uniquely as the dot product with a vector:
this vector is the cross product \scriptstyle a \times b. From this
perspective, the cross product is
defined by the
scalar triple product,
\scriptstyle\mathrm{Vol}(a,b,c) = (a\times b)\cdot c.
In the same way, in higher dimensions one may define generalized
cross products by raising indices of the
n-dimensional
volume form, which is a \scriptstyle (0,n)-tensor.The most direct
generalizations of the cross product are to define either:
- a \scriptstyle (1,n-1)-tensor, which takes as input
\scriptstyle n-1 vectors, and gives as output 1 vector – an
\scriptstyle (n-1)-ary vector-valued product, or
- a \scriptstyle (n-2,2)-tensor, which takes as input 2 vectors
and gives as output skew-symmetric
tensor of rank n−2 – a binary product with rank
n−2 tensor values. One can also define \scriptstyle
(k,n-k)-tensors for other k.
These products are all multilinear and skew-symmetric, and can be
defined in terms of the determinant and
parity.
The \scriptstyle (n-1)-ary product can be described as follows:
given \scriptstyle n-1 vectors \scriptstyle v_1,\dots,v_{n-1} in
\scriptstyle\mathbf{R}^n, define their generalized cross product
\scriptstyle v_n = v_1 \times \cdots \times v_{n-1} as:
- perpendicular to the hyperplane defined by the \scriptstyle
v_i,
- magnitude is the volume of the parallelotope defined by the \scriptstyle v_i,
which can be computed as the Gram
determinant of the \scriptstyle v_i,
- oriented so that \scriptstyle v_1,\dots,v_n is positively
oriented.
This is the unique multilinear, alternating product which evaluates
to \scriptstyle e_1 \times \dots \times e_{n-1} = e_n, \scriptstyle
e_2 \times \dots \times e_n = e_1, and so forth for cyclic
permutations of indices.
In coordinates, one can give a formula for this
n-ary
analogue of the cross product in
R^{n+1} by:
\bigwedge(\mathbf{v}_1,\dots,\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
R^{3}
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
(
v_{1},...,
v_{n},Λ(
v_{1},...,
v_{n}))
have a positive
orientation with respect to
(
e_{1},...,
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.
History
In 1773,
Joseph Louis Lagrange
introduced the component form of both the dot and cross products in
order to study the
tetrahedron in three
dimensions. In 1843 the Irish mathematical physicist Sir
William Rowan Hamilton introduced the
quaternion product, and with it the terms
"vector" and "scalar". Given two quaternions [0,
u] and [0,
v], where
u and
v are vectors in
R^{3}, their quaternion product can be
summarized as [−
u·
v,
u×
v].
James Clerk Maxwell used Hamilton's
quaternion tools to develop his famous
electromagnetism equations, and for this
and other reasons quaternions for a time were an essential part of
physics education.
However,
Oliver Heaviside in England and Josiah Willard Gibbs in Connecticut felt that quaternion methods were too cumbersome,
often requiring the scalar or vector part of a result to be
extracted. Thus, about forty years after the quaternion
product, the
dot product and cross
product were introduced—to heated opposition. Pivotal to (eventual)
acceptance was the efficiency of the new approach, allowing
Heaviside to reduce the equations of electromagnetism from
Maxwell's original 20 to the four commonly seen today.
Largely independent of this development, and largely unappreciated
at the time,
Hermann Grassmann
created a geometric algebra not tied to dimension two or three,
with the
exterior product playing a
central role.
William Kingdon
Clifford combined the algebras of Hamilton and Grassmann to
produce
Clifford algebra, where in
the case of three-dimensional vectors the bivector produced from
two vectors dualizes to a vector, thus reproducing the cross
product.
The cross notation, which began with Gibbs, inspired the name
"cross product". Originally appearing in privately published notes
for his students in 1881 as
Elements of Vector Analysis,
Gibbs's notation—and the name—later reached a wider audience
through
Vector
Analysis , a textbook by a former student.
Edwin Bidwell Wilson rearranged
material from Gibbs's lectures, together with material from
publications by Heaviside, Föpps, and Hamilton. He divided
vector analysis into three parts:
Two main kinds of vector multiplications were defined, and they
were called as follows:
- The direct, scalar, or
dot product of two vectors
- The skew, vector, or
cross product of two vectors
Several kinds of
triple products and
products of more than three vectors were also examined. The above
mentioned triple product expansion was also included.
See also
Notes
Other notes
References
External links