When it comes to geometry and mathematics, numerous terms often seem to mean the same thing but actually, that is not the case! The same is the case of a perpendicular pair and an orthogonal figure. And thus, this article will help you understand what these two terms mean and what are the fine differences between them. With the help of descriptive pointers and comparison tables, this article will make sure to leave no doubts uncleared on the way to understand the perpendicular and an orthogonal pair.
Perpendicular vs Orthogonal
The main difference between perpendicular and orthogonal is that perpendicular is a phenomenon and it means that a straight line makes a right angle to another line which can never be parallel. The term talks about the ninety-degree angle and the relationship between the two lines whereas, the term orthogonal is rather a condition and a positioning i.e. it describes the relationship between the two lines with respect to each other and not only the angle between them. Let’s talk further about their definition to get more clarity.
Perpendicular paths are two separate lines that meet at a 90-degree angle. Have you observed something similar to the symbol “L” or the connecting points of your wall surfaces? They are perpendicular planes, which are straight lines forming two planes that meet at a certain degree – the right angle. “When two planes or lines meet at a 90° angle, we say they are perpendicular”
Now, as mentioned earlier. The phenomenon of this occurrence and this situation where a right angle is formed while the lines are not being parallel to each other is named as a perpendicular.
Talking about the orthogonal relationship or orthogonality; it is a mathematical concept that extends the concept of role orientations to the linear algebra of piecewise linear forms and the definition of how a perpendicular pair exists. When B(u, v) = 0, two components u and v of a subspace with the bilinear prescribed format are orthogonal. The vector field may include non-zero self-orthogonal variables based on the bilinear form. Groups of properly functioning are used to build a basis in which values are distributed.
Comparison Table Between Perpendicular and Orthogonal
| Parameters of Comparison | Perpendicular | Orthogonal |
| Meaning (Geometric) | Perpendicular paths are two separate lines that meet at a 90-degree angle. | Orthogonality, when extended to matrices, this feature is equivalent with perpendicularity, although it also applies to functional aspects more broadly. |
| Relationship | 1. If two lines meet, one first line is “perpendicular” to the second and vice versa. 2. At the point of incidence, the straight (180) angle on one end of the very first line is split into two corresponding angles by the second plane making them perpendicular as well as orthogonally positive. | 1. The property and functional aspect of an orthogonal pair is similar to a perpendicular. 2. The dot product of two vector components of an orthogonal pair is zero. |
| Statistical Relation | The two lines are statistically dependent and the angles are non-constant if either is changed. | The two components of an orthogonal pair are statistically independent of each other. |
| Terminology | Logical and geometric terminology. | Mathematical and geometrical terminology with regards to vector physics. |
| Etymology | From the old French and Latin word ‘perpendicularis’ meaning vertical to the plane. | Late 16th century: from French, based on Greek orthogōnios ‘right-angled’. |
What is Perpendicular?
When two lines or planes cross at a right angle forming angle, the two lines are seen as being perpendicular to each other. Explicitly, if two lines meet, one first line is “orthogonal” to the second; and secondly, at the point of incidence, the straight (180) angle on one end of the very first line is split into two corresponding angles by the second plane making them perpendiculars as well as orthogonally positive.
Perpendicularity is symmetrical, which means that if one line is perpendicular to another, the second line is likewise equally perpendicular to the first. As a result, we may refer to two planes and lines as perpendicular (to each other) without mentioning their sequence.
The idea and existence of perpendicular line segments have already been demonstrated. The equivalent angle at the vertices of an “L” form in a figure is “always” a right angle. All crossing planes or lines are perpendicular to each other, but not all meeting lines are perpendicular to one another. Perpendicular lines have two primary characteristics:
- Lines that are perpendicular to one other usually meet or cross.
- Any angle formed by two line segments that are claimed to be perpendicular is always 90 degrees.
Do not confuse perpendiculars with “parallels” as they are two straight lines that are separated from one other and never intersect, regardless of how far in either side they are however, perpendiculars even if stretched till infinity, always intersect or rather “cross” each other.
Parallel pairings could never be regarded as a perpendicular pair, and they can never be orthogonally positive. Room wall intersection points, the sides of a cube and a cuboid are all perpendicular to each other, and a tree standing straight vertically is perpendicular to the earth‘s surface are all instances of perpendiculars. Two perpendicular lines are represented by the symbol: ⊥.
What is Orthogonal?
Orthogonality, when extended to matrices, this feature is equivalent to perpendicularity, although it also applies to functional aspects more broadly. When the partial derivative is a vector, the dot product (see vector operations); for functions, the definite integral of their multiplication—is 0, two components of an n-dimensional space are always orthogonal. In geometry, it is simply a property that superimposes the properties of a perpendicular pair; it is often used in the determination of two congruent triangles.
An inner product structure may be produced from a concatenation of the components of a set of perpendicular vectors or functions, which means that any component of the space can be generated from the members of such a set.
Orthogonality, When extended to matrices, this feature is equivalent with perpendicularity, although it also applies to functional aspects more broadly. When the partial derivative is a vector, the dot product (see vector operations); for functions, the definite integral of their multiplication—is 0, two components of an n-dimensional space are always orthogonal.
An inner product structure may be produced from a concatenation of the components of a set of perpendicular vectors or functions, which means that any component of the space can be generated from the members of such a set.
Main Differences Between Perpendicular and Orthogonal
- Perpendicular is a condition whereas orthogonality is a property.
- Perpendicular pairs are statistically dependent whereas orthogonal pairs are statistically independent.
- Perpendicular terminology is referred to in geometry only whereas orthogonal or orthogonality is a part of geometry as well as vector physics where it correlates to the vector functionalities as an independent property.
- Perpendicular also means vertical position whereas other meanings of orthogonal include; “of two or more conditions in a single problem”.
- Perpendicular is more suitable in describing the positioning of an object whereas the “orthogonal” term is used to mathematically prove the same condition.
Conclusion
Two vectors are orthogonal if or unless their dot product is always equal to zero, i.e. they create an aspect of 90°, or one of the vectors is zero, according to the Euclidean plane hypothesis. As a result, orthogonality of vector pairs is a generalization of the idea of perpendicular lines to any degree of space. Perpendicular is a word that is commonly used in both mathematics and everyday life.
Both terminologies are linked by the fact that their components are oriented at a right angle to one another. The orthogonality characteristics, on the other hand, have a different meaning and are incongruent in the case of the vector dot product concept.
References
- https://www.britannica.com/science/orthogonality
- https://www.cuemath.com/geometry/perpendicular/