Editing Cross product (section)

=== Alternative formulation ===
The cross product and the dot product are related by:
:<math> \left\| \mathbf{a} \times \mathbf{b} \right\| ^2  = \left\| \mathbf{a}\right\|^2  \left\|\mathbf{b}\right\|^2 - (\mathbf{a} \cdot \mathbf{b})^2 .</math>

The right-hand side is the [[Gramian matrix|Gram determinant]] of '''a''' and '''b''', the square of the area of the parallelogram defined by the vectors. This condition determines the magnitude of the cross product. Namely, since the dot product is defined, in terms of the angle ''θ'' between the two vectors, as:

:<math> \mathbf{a \cdot b} = \left\| \mathbf a \right\| \left\| \mathbf b \right\| \cos \theta , </math>

the above given relationship can be rewritten as follows:

:<math>  \left\| \mathbf{a \times b} \right\|^2 = \left\| \mathbf{a} \right\| ^2 \left\| \mathbf{b}\right \| ^2 \left(1-\cos^2 \theta \right) .</math>

Invoking the [[Pythagorean trigonometric identity]] one obtains:
:<math> \left\| \mathbf{a} \times \mathbf{b} \right\| = \left\| \mathbf{a} \right\| \left\| \mathbf{b} \right\| \left| \sin \theta \right| ,</math>

which is the magnitude of the cross product expressed in terms of ''θ'', equal to the area of the parallelogram defined by '''a''' and '''b''' (see [[#Definition|definition]] above).

The combination of this requirement and the property that the cross product be orthogonal to its constituents '''a''' and '''b''' provides an alternative definition of the cross product.<ref name=Massey>{{cite journal |title=Cross products of vectors in higher dimensional Euclidean spaces |author=WS Massey |journal=The American Mathematical Monthly |volume=90 |date=Dec 1983 |pages=697–701 |issue=10 |doi=10.2307/2323537 |publisher=The American Mathematical Monthly, Vol. 90, No. 10 |jstor=2323537}}</ref>