[nb 1][1] Most of these relations can be dated to founder of vector calculus Josiah Willard Gibbs, if not earlier.
[2] The magnitude of a vector A can be expressed using the dot product: In three-dimensional Euclidean space, the magnitude of a vector is determined from its three components using Pythagoras' theorem: The vector product and the scalar product of two vectors define the angle between them, say θ:[1][3] To satisfy the right-hand rule, for positive θ, vector B is counter-clockwise from A, and for negative θ it is clockwise.
The Pythagorean trigonometric identity then provides: If a vector A = (Ax, Ay, Az) makes angles α, β, γ with an orthogonal set of x-, y- and z-axes, then: and analogously for angles β, γ. Consequently: with
The area Σ of a parallelogram with sides A and B containing the angle θ is: which will be recognized as the magnitude of the vector cross product of the vectors A and B lying along the sides of the parallelogram.
That is: (If A, B are two-dimensional vectors, this is equal to the determinant of the 2 × 2 matrix with rows A, B.)
The square of this expression is:[4] where Γ(A, B) is the Gram determinant of A and B defined by: In a similar fashion, the squared volume V of a parallelepiped spanned by the three vectors A, B, C is given by the Gram determinant of the three vectors:[4] Since A, B, C are three-dimensional vectors, this is equal to the square of the scalar triple product
[2] It can be evaluated using the identity:[7] using the notation for the triple product: Equivalent forms can be obtained using the identity:[8][9][10] This identity can also be written using tensor notation and the Einstein summation convention as follows: where εijk is the Levi-Civita symbol.
For example, if four points are chosen on the unit sphere, A, B, C, D, and unit vectors drawn from the center of the sphere to the four points, a, b, c, d respectively, the identity: in conjunction with the relation for the magnitude of the cross product: and the dot product: where a = b = 1 for the unit sphere, results in the identity among the angles attributed to Gauss: where x is the angle between a × b and c × d, or equivalently, between the planes defined by these vectors.