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In mathematics, the concept of orthogonal plays a vital role. It has several meanings. Basically, the orthogonal refers to the perpendicular. Etymologically, the word "orthogonal" was originated from the composition of two Greek words -"orthos" and "gonia". The former means "straight or upright", while the latter referring to the "angle". In ancient Greek, orthogonium originally represented a rectangle. Later, the word orthogonal was utilized in order to denote a right angle or something related to a right angle. The phenomenon of being orthogonal is known as orthogonality.

Presently, the term orthogonal or orthogonality utilized differently in different contexts. In this page below, we shall go ahead and understand about the concept of orthogonality and its definitions in reference to different mathematical areas.


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When we define orthogonal or orthogonality, we find several definitions in various contexts. Few important and most frequently used ones are illustrated below:

1) In the reference of geometry, two lines are called orthogonal if the lines are at right angles to each other. The symbol $\perp$ is used to denote this orthogonality.

2) In linear algebra, two vectors are said to be orthogonal to each other, if their inner product vanishes. In other words, the vectors x and y are orthogonal if the inner product <x , y> is equal to zero. This is represented by x $\perp$ y.

3) Two curves in a plane are orthogonal at an intersection point if the tangent lines at that point are perpendicular.

4) Let us suppose a linear transformation T : A $\rightarrow$ A. It is termed as an orthogonal linear transformation, if inner product is preserved. Thus, for every possible pair of vectors a and b, <T$_{a}$, T$_{b}$> = <a, b>.

Orthogonal Basis

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Linear algebra is an important branch of mathematics that deals with vectors and linear equations. In this branch, a term is "basis" is used very commonly. A basis is defined as a set containing vectors that are linearly independent. Vector in a basis are to be represented in a linear combination. In other words, we can understand a basis  as a linearly independent span set.

In linear algebra, there is a useful concept of orthogonal basis that is defined as a type of basis whose elements or vectors of which are mutually perpendicular or orthogonal to each other. That is, for each pair of vectors in an orthogonal basis, the dot product is equal to zero. Let us consider that there are two vectors 'a' and 'b' belonging to orthogonal basis X. Then,
a . b = 0 , where a , b $\in$ X

How to Find Orthogonal Basis?

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In order to find orthogonal basis, the students are required to follow a certain algorithm. The most common method is Gram Schmidt algorithm. Based on this method, the procedure of determining an orthonormal basis for a given inner product space is described as follows:
Let us consider that W be an inner product space and X = {$x_{1},x_{2},x_{3},...,x_{n}$} be a basis of W. Then, the steps to be followed are:

Step 1: Define $y_{1}=x_{1}$.

Step 2: Now compute the value of $y_{i}$ for each 2 $\leq$ i $\leq$ n by using the relation:

$y_{i}$=$x_{i}-u_{1}$ $\frac{<x_{i}\ ,\ y_{1}>}{<y_{1}\ ,\ y_{1}>}$ - $u_{2}$ $\frac{<x_{i}\ ,\ y_{2}>}{<y_{2}\ ,\ y_{2}>}$ -...-$u_{i-1}$ $\frac{<x_{i}\ ,\ y_{i-1}>}{<y_{i-1}\ ,\ y_{i-1}>}$.

Step 3: Define $x_{i}$=$\frac{y_{i}}{|y_{i}|}$; where, 1 $\leq$ i $\leq$ n
X = {$x_{1},x_{2},x_{3},...,x_{n}$} is the required orthogonal basis of W.

Orthogonal Functions

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In mathematics, a function is defined as a relation in which every permissible input value does have one single output value. Orthogonal functions are used frequently in maths. Two functions are said to be orthogonal functions when their inner product is zero. According to the formal definition of orthogonal functions:

The two functions f(x) and g(x) are termed as orthogonal functions if <f(x) , g(x)> i.e. their inner product is equal to zero, provided that f(x) $\neq$ g(x). For f(x) and g(x) to be orthogonal over the interval p $\leq$ x $\leq$ q.

<f(x) , g(x)> = $\int_{p}^{q}f(x)\ g(x)\ wt(x)$ = 0

Where, wt(x) is the weight function.

Orthogonal Lines

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The most common definition of orthogonality is used in context with geometry. In geometry, we learn about lines and shapes. We come across with orthogonal lines very frequently. Orthogonal lines are the lines that are perpendicular to each other. More elaborately, two lines or line segments situated in the same plane are said to be orthogonal lines if one makes a right angle with another. If these lines cross each other, they make four right angles at the point of intersection. Such lines are known as "orthogonal to each other".

In short, we can say that the orthogonal lines are mutually perpendicular. The following image demonstrates two orthogonal lines.

Orthogonal Lines

Orthogonal Vectors

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Orthogonal vectors play a significant role in linear or vector algebra. The two vectors are said to be orthogonal vectors if their dot product is equal to zero. Let us suppose that there are two vectors A and B belonging to a vector space X. If -
$\vec{A}\ .\ \vec{B}$ = 0

then $\vec{A}$ and $\vec{B}$ are called orthogonal vectors. These are also termed as mutually perpendicular vectors. This relation is denoted by $\vec{A} \perp \vec{B}$

For Example:
$\vec{A}$ = (2, 4) and $\vec{A}$ = {2, -1} are orthogonal vectors; since

$\vec{A}.\vec{B}$ = 2 x 2 + 4 x (-1)

= 4 - 4 = 0

Orthogonal View

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Orthogonal view is an important concept. An 'orthogonal view' is a view of a three dimensional object in a two dimensional plane. Orthogonal view is considered as a kind of parallel projection. Orthogonal view is the tracing of three dimensional figures on a two dimensional paper from different angles. It was used in ancient times for cartographic purposes. In an orthogonal view, all the projecting lines are perpendicular or orthogonal to the two dimensional projection plane.

A diagram illustrating orthogonal view is given below:
Orthogonal View
Orthogonal view of objects is applied to many areas in mathematics as well as in various fields beyond mathematics.

For Example: architecture, animation, cartography, orthographic projection etc.