# Cross product – 벡터 곱

수학에서 외적(外積)은 3차원 공간의 벡터들간의 이항연산의 일종이다.
연산의 결과가 스칼라인 내적(內積)과는 달리 연산의 결과가 벡터이기 때문에 벡터곱(vector product)이라고
불리기도 한다. 외적은 물리학의 각운동량, 로렌츠 힘등의 공식에 등장한다.

정의

두 벡터 ab의 외적은 a x b라 쓰고(수학자들은 a ^ b라고 쓰기도 한다.), 다음과 같이 정의된다.

$\mathbf{a} \times \mathbf{b} = \mathbf\hat{n} \left| \mathbf{a} \right| \left| \mathbf{b} \right| \sin \theta$

식에서 θ는 ab가 이루는 각을 나타내며, nab에 공통으로 수직인 단위벡터를 나타낸다.

위 정의에서의 문제점은 ab에 공통으로 수직인 방향이 두개라는 점이다. 즉, n이 수직이면, -n도 수직이다.

어느 것을 두 벡터의 외적으로 할 것인가는 벡터공간의 방향성(orientation)에 따라 달라진다.
오른손 좌표계에서는 a×b는, a, b, a×b가 오른손 좌표계 방향을 따르도록 정의되고, 왼손좌표계에선
마찬가지로 이 순서의 세 벡터가 왼손 좌표계 방향을 따르도록 정의된다.

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In mathematics, the cross product is a binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which is perpendicular to the two input vectors. By contrast, the dot product 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-dimensions (and the algebra defined by the cross product is not associative). 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 a × b. In a three-dimensional Euclidean space, with a usual right-handed coordinate system, it 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 given by the formula

$\mathbf{a} \times \mathbf{b} = a b \sin \theta \ \mathbf{\hat{n}}$

where θ is the measure of the (non-obtuse) angle between a and b (0° ≤ θ ≤ 180°), a and b are the magnitudes of vectors a and b, and $\mathbf{\hat{n}}$ 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 $\mathbf{\hat{n}}$ 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 $\mathbf{\hat{n}}$ is coming out of the thumb (see the picture on the right).

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 $\mathbf{\hat{n}}$ 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 $\mathbf{\hat{n}}$. 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.