Vectors in n-dimensional space. Orthogonal vectors. Pythagorean theorem. Orthogonal and orthonormal bases.

Length of a vector in n-dimensional space. If x is a vector in n-space, its length is given by

i.e. the square root of the dot product x•x.

Distance between points in n-space. If x and y represent two points in n-space the distance between them is given by

(i.e. the length of the vector x - y )

Angle between two vectors in n-space. If x and y are two vectors in n-space the angle θ between them is given by

This formula is a direct generalization of the formula x•y = ||x|| ||y|| cos θ for three dimensional space. By the Schwarz inequality (theorem),

            |x•y| ||x|| ||y||

for any two n-vectors x and y. Thus the fraction

is always in the range -1 to +1.

Orthogonal vectors in n-space. Two vectors x and y in n-space are said to be orthogonal if and only if

(i.e. they are orthogonal if cos θ = 0, and thus θ = 90^o.)

Pythagorean theorem in n-space. Let x₁, x₂, ...... , x_k be k pairwise orthogonal vectors in n-space. Then

            ( x₁ + x₂ + ...... + x_k ) · ( x₁ + x₂ + ...... + x_k ) = x₁ · x₁ + x₂ · x₂ + ...... + x_k · x_k

i.e. the square of the diagonal of the k-dimensional rectangular parallelepiped formed by x₁, x₂, ...... , x_k is equal to the sum of the squares of the edges. See Fig. 1 for case k = 3.

Proof

Basis for an n-dimensional space. A basis for an n-dimensional space is any set of linearly independent vectors that span the space. Every vector in the space the can be expressed as a unique linear combination of the vectors in this basis.

Orthogonal basis for an n-dimensional space. An orthogonal basis for an n-dimensional space is any set of n pairwise orthogonal vectors in the space. Every vector in the space the can be expressed as a unique linear combination of the vectors in this basis.

Orthonormal basis for an n-dimensional space. An orthonormal basis for an n-dimensional space is any set of n pairwise orthogonal unit vectors in the space. If x₁, x₂, ...... , x_n are a set of n pairwise orthogonal vectors in the space, than the set

constitutes an orthonormal basis for the space. Every vector in the space the can be expressed as a unique linear combination of the vectors in this basis.

Vectors referred to an orthonormal basis. Let the vectors x₁, x₂, ...... , x_n represent an orthonormal basis for an n-dimensional space. Then any vector x in the space can be represented as

            x = c₁x₁ + c₂x₂ + .......... + c_nx_n

where c₁, c₂, ...... , c_n are constants. The coefficients c₁, c₂, ...... , c_n represent the projections of the vector x on the basis vectors x₁, x₂, ...... , x_n (they can be viewed as the coordinates of point x in the orthonormal system).