Vector algebra relations

The following are important identities in vector algebra. Identities that only involve the magnitude of a vector ${\displaystyle \|\mathbf {A} \|}$ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there. Most of these relations can be dated to founder of vector calculus Josiah Willard Gibbs, if not earlier.

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:

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 = (A_x, A_y, A_z) makes angles α, β, γ with an orthogonal set of x-, y- and z-axes, then:

and analogously for angles β, γ. Consequently: