In mathematics, the cross product is a binary operation on vectorss in three dimensions. 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.
Table of contents |
2 Properties 3 Applications 4 Higher dimensions |
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 is defined as:
The problem with this definition is that there are two unit vectors perpendicular to both a and b: if n is perpedicular, 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. In a right-handed system a×b is defined so that a, b and a×b also becomes a right handed system. If i, j, k is left-handed, then a, b and a×b is defined to be left-handed. 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:
Properties
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.The cross product is anti-symmetric, which means:
- a×b = -b×a
- a×(b + c) = a×b + a×c
- (ra)×b = a×(rb) = r(a×b).
The cross product is not associative, but satisfies the Jacobi identity:
- a×(b×c) + b×(c×a) + c×(a×b) = 0
- a×(b×c) = (a·c)b - (a·b)c
- i×j = k, j×k = i, k×i = j
- a×b = [a_{2}b_{3} - a_{3}b_{2}, a_{3}b_{1} - a_{1}b_{3}, a_{1}b_{2} - a_{2}b_{1}]
- det(a,b,c) = a·(b×c).
Applications
The cross product occurs in the formula for the vector operator curl. The cross product is also used to describe the Lorenz 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:
- It is bilinear in the sense that x×(ay+bz) = ax×y+bx×z and (ay+bz)×x = ay×x+bz×x
- It is anti-commutative: 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
- We have ||x×y||^{2} = ||x||^{2}||y||^{2}-(x·y)^{2}
See also: Right-handed rule