** Orthogonal vs Orthonormal
**

In mathematics, the two words orthogonal and orthonormal are frequently used along with a set of vectors. Here, the term ‘vector’ is used in the sense that it is an element of a vector space – an algebraic structure used in linear algebra. For our discussion, we will consider an inner-product space – a vector space **V** along with an inner product **[ ]** defined on **V**.

As an example, for an inner product, space is the set of all 3-dimensional position vectors along with the usual dot product.

**What is orthogonal?**

A nonempty subset **S **of an inner product space **V **is said to be orthogonal, if and only if for each *distinct ***u, v **in **S**, **[u, v] = **0; i.e. the inner product of **u **and **v **is equal to the zero scalar in the inner product space.

For example, in the set of all 3-dimensional position vectors, this is equivalent to saying that, for each distinct pair of position vectors **p **and** q** in S, **p **and** q **are perpendicular to each other. (Remember that the inner product in this vector space is the dot product. Also, the dot product of two vectors is equal to 0 if and only if the two vectors are perpendicular to each other.)

Consider the set **S** = {(0,2,0), (4,0,0), (0,0,5)}, which is a subset of the 3-dimensional position vectors. Observe that (0,2,0).(4,0,0) = 0**, **(4,0,0)**.**(0,0,5) = 0 & (0,2,0)**.**(0,0,5) = 0. Hence, the set **S **is orthogonal. In particular, two vectors are said to be orthogonal if their inner product is 0. Therefore, each pair of vectors in** S**is orthogonal.

**What is orthonormal?**

A nonempty subset **S** of an inner product space **V** is said to be orthonormal if and only if **S **is orthogonal and for each vector **u **in **S**, **[u, u] = **1. Therefore, it can be seen that every orthonormal set is orthogonal but not vice versa.

For example, in the set of all 3-dimensional position vectors, this is equivalent to saying that, for each distinct pair of position vectors **p **and** q** in **S**, **p **and** q** are perpendicular to each other, and for each **p **in** S**, **|p| = **1. This is because the condition **[p, p] = **1 reduces to **p.p=|p||p|**cos0 = **|p|**^{2}=1, which is equivalent to **|p| = **1. Therefore, given an orthogonal set we can always form a corresponding orthonormal set by dividing each vector by its magnitude.

**T** = {(0,1,0), (1,0,0), (0,0,1)} is an orthonormal subset of the set of all 3-dimensional position vectors. It is easy to see that it was obtained by dividing each of the vectors in the set **S**, by their magnitudes.

**What is the difference between orthogonal and orthonormal?**

- A nonempty subset
**S**of an inner product space**V**is said to be orthogonal, if and only if for each distinct**u, v**in**S**,**[u, v] =**0. However, it is orthonormal, if and only if an additional condition – for each vector**u**in**S**,**[u, u] =**1 is satisfied. - Any orthonormal set is orthogonal but not vice-versa.
- Any orthogonal set corresponds to a unique orthonormal set but an orthonormal set may correspond to many orthogonal sets.

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