In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal unit vectors. A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the two vectors are perpendicular to each other. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis.
Intuitive overview The construction of orthogonality of vectors is motivated by a desire to extend the intuitive notion of perpendicular vectors to higher-dimensional spaces. In the Cartesian plane, two vectors are said to be perpendicular if the angle between them is 90° (i.e. if they form a right angle). This definition can be formalized in Cartesian space by defining the dot product and specifying that two vectors in the plane are orthogonal if their dot product is zero. Similarly, the construction of the norm of a vector is motivated by a desire to extend the intuitive notion of the length of a vector to higher-dimensional spaces. In Cartesian space, the norm of a vector is the square root of the vector dotted with itself. That is,
‖ x ‖ = x ⋅ x {\displaystyle \|\mathbf {x} \|={\sqrt {\mathbf {x} \cdot \mathbf {x} }}}
Many important results in linear algebra deal with collections of two or more orthogonal vectors. But often, it is easier to deal with vectors of unit length. That is, it often simplifies things to only consider vectors whose norm equals 1. The notion of restricting orthogonal pairs of vectors to only those of unit length is important enough to be given a special name. Two vectors which are orthogonal and of length 1 are said to be orthonormal.
Simple example What does a pair of orthonormal vectors in 2-D Euclidean space look like? Let u = (x1, y1) and v = (x2, y2). Consider the restrictions on x1, x2, y1, y2 required to make u and v form an orthonormal pair.
From the orthogonality restriction, u • v = 0. From the unit length restriction on u, ||u|| = 1. From the unit length restriction on v, ||v|| = 1. Expanding these terms gives 3 equations:
x 1 x 2 + y 1 y 2 = 0 {\displaystyle x_{1}x_{2}+y_{1}y_{2}=0\quad }
x 1 2 + y 1 2 = 1 {\displaystyle {\sqrt {{x_{1}}^{2}+{y_{1}}^{2}}}=1}
x 2 2 + y 2 2 = 1 {\displaystyle {\sqrt {{x_{2}}^{2}+{y_{2}}^{2}}}=1}
Converting from Cartesian to polar coordinates, and considering Equation ( 2 ) {\displaystyle (2)} and Equation ( 3 ) {\displaystyle (3)} immediately gives the result r1 = r2 = 1. In other words, requiring the vectors be of unit length restricts the vectors to lie on the unit circle. After substitution, Equation ( 1 ) {\displaystyle (1)} becomes cos θ 1 cos θ 2 + sin θ 1 sin θ 2 = 0 {\displaystyle \cos \theta _{1}\cos \theta _{2}+\sin \theta _{1}\sin \theta _{2}=0} . Rearranging gives tan θ 1 = − cot θ 2 {\displaystyle \tan \theta _{1}=-\cot \theta _{2}} . Using a trigonometric identity to convert the cotangent term gives
tan ( θ 1 ) = tan ( θ 2 + π 2 ) {\displaystyle \tan(\theta _{1})=\tan \left(\theta _{2}+{\tfrac {\pi }{2}}\right)}
⇒ θ 1 = θ 2 + π 2 {\displaystyle \Rightarrow \theta _{1}=\theta _{2}+{\tfrac {\pi }{2}}}
It is clear that in the plane, orthonormal vectors are simply radii of the unit circle whose difference in angles equals 90°.
… excerpt ends here. Continue reading the full article.
