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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 a
Del operator.
A special case, useful in vector calculus, is
-
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:
-
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
-
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:
-
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.
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:
- x × (ay + bz) = ax × y +
bx × z
- (ay + bz) × x = ay × x +
bz × x.
- 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
- |x × y|2 = |x|2 |y|2
− (x · y)2.
See also: Right-handed rule, evaluating cross products
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