Editing Euclidean vector (section)

===Cross product===
{{main|Cross product}}

The ''cross product'' (also called the ''vector product'' or ''outer product'') is only meaningful in three or [[Seven-dimensional cross product|seven]] dimensions. The cross product differs from the dot product primarily in that the result of the cross product of two vectors is a vector. The cross product, denoted '''a'''&nbsp;×&nbsp;'''b''', is a vector perpendicular to both '''a''' and '''b''' and is defined as

<math display=block>\mathbf{a}\times\mathbf{b}
=\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\sin(\theta)\,\mathbf{n}</math>

where ''θ'' is the measure of the angle between '''a''' and '''b''', and '''n''' is a unit vector [[perpendicular]] to both '''a''' and '''b''' which completes a [[Right-hand rule|right-handed]] system. The right-handedness constraint is necessary because there exist ''two'' unit vectors that are perpendicular to both '''a''' and '''b''', namely, '''n''' and (−'''n''').
[[Image:Cross product vector.svg|class=skin-invert-image|thumb|right|An illustration of the cross product]]

The cross product '''a'''&nbsp;×&nbsp;'''b''' is defined so that '''a''', '''b''', and '''a'''&nbsp;×&nbsp;'''b''' also becomes a right-handed system (although '''a''' and '''b''' are not necessarily [[orthogonal]]). This is the [[right-hand rule]].

The length of '''a'''&nbsp;×&nbsp;'''b''' can be interpreted as the area of the parallelogram having '''a''' and '''b''' as sides.

The cross product can be written as
<math display=block>{\mathbf a}\times{\mathbf b} = (a_2 b_3 - a_3 b_2) {\mathbf e}_1 + (a_3 b_1 - a_1 b_3) {\mathbf e}_2 + (a_1 b_2 - a_2 b_1) {\mathbf e}_3.</math>

For arbitrary choices of spatial orientation (that is, allowing for left-handed as well as right-handed coordinate systems) the cross product of two vectors is a [[pseudovector]] instead of a vector (see below).