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.
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4 Higher dimensions
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:
where θ is the measure of the angle between a and b (between 0 and 180 degrees, or between 0 and π if measured in radian), and n is a unit vector perpendicular to both a and b.
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:
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:
The unit vectors i, j, k from the given orthogonal coordinate system satisfy:
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.
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: