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Cross Product
In mathematics, the cross product is a binary operation on two vectors in a threedimensional 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 associative nor commutative. It contrasts with the dot product which produces a
scalar result. In many engineering and physics problems, it is handy 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 known as the vector product, or
Gibbs vector product.
The cross product is not defined except 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.
The cross product of two vectors a and b is denoted by a × b. In physics, sometimes the
notation a ∧ b is used (mathematicians do not use this notation, 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
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 is a unit vector perpendicular to the plane
containing a and b. 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 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 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 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 lefthanded coordinate system), should not change the direction of . The problem is clarified
by realizing that the cross-product of two vectors is not a (true) vector, but rather a
pseudovector.
The magnitude of the cross product can be interpreted as the positive area of the
parallelogram having a and b as sides:
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):
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:
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 R3 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(3).
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.