Orthonormals

Orthonormals are a set of vectors that are both orthogonal (perpendicular) and normalized (unit length) to each other. In a Euclidean space, orthonormal vectors form a basis for the space, meaning any vector in the space can be expressed as a linear combination of these basis vectors. This property makes orthonormal vectors particularly useful in various mathematical and scientific applications, such as signal processing, quantum mechanics, and computer graphics.

In a two-dimensional space, orthonormal vectors can be represented as i and j, where i = (1, 0) and j = (0, 1). In a three-dimensional space, orthonormal vectors can be represented as i, j, and k, where i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1). These vectors are often referred to as the standard basis vectors.

Orthonormal vectors have several useful properties. First, the dot product of any two distinct orthonormal vectors is zero, which is a direct consequence of their orthogonality. Second, the dot product of any orthonormal vector with itself is one, which is a direct consequence of their normalization. These properties make orthonormal vectors particularly useful in various mathematical and scientific applications.

Orthonormal vectors can be generated from any set of linearly independent vectors using the Gram-Schmidt process. This process involves iteratively projecting each vector onto the subspace spanned by the previously generated orthonormal vectors and then normalizing the resulting vector. The resulting set of orthonormal vectors is said to be orthonormal with respect to the original set of linearly independent vectors.