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.

Magnitudes

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:

Inequalities

Here the notation denotes the dot product of vectors A and B.

Angles

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 = makes angles α, β, γ with an orthogonal set of x-, y- and z-axes, then:
and analogously for angles β, γ. Consequently:
with unit vectors along the axis directions.

Areas and volumes

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:
The square of this expression is:
where Γ 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:
Since A, B, C are three-dimensional vectors, this is equal to the square of the scalar triple product below.
This process can be extended to n-dimensions.

Addition and multiplication of vectors

Some of the following algebraic relations refer to the dot product and the cross product of vectors.