Vector algebra relations

The relations below apply to vectors in a three-dimensional Euclidean space. Some, but not all of them, extend to vectors of higher dimensions. In particular, the cross product of vectors is defined only in three dimensions (but see Seven-dimensional cross product). The relations below apply to vectors in a three-dimensional Euclidean space. Some, but not all of them, extend to vectors of higher dimensions. In particular, the cross product of vectors is defined only in three dimensions (but see Seven-dimensional cross product). The magnitude of a vector A is determined by its three components along three orthogonal directions using Pythagoras' theorem: The magnitude also can be expressed using the dot product: Here the notation (A · B) denotes the dot product of vectors A and B. 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. Here the notation A × B denotes the vector cross product of vectors A and B.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: