Vector algebra relations

The following are important identities in vector algebra. Identities that involve the magnitude of a vector formula_1, or the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension. Identities that use the cross product (vector product) A×B are defined only in three dimensions. (There is a seven-dimensional cross product, but the identities do not hold in seven dimensions.)
Magnitudes.
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:
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.
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 formula_12 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:
(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:
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:
Since A, B, C are three-dimensional vectors, this is equal to the square of the scalar triple product formula_18 below.
This process can be extended to "n"-dimensions.