# Perpendicular vs. Orthogonal

By Jaxson

## Main Difference

The main difference between Perpendicular and Orthogonal is that the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees). The property extends to other related geometric objects and Orthogonal is a relation of two lines at right angles.

• Perpendicular

In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees). The property extends to other related geometric objects.

A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the straight angle on one side of the first line is cut by the second line into two congruent angles. Perpendicularity can be shown to be symmetric, meaning if a first line is perpendicular to a second line, then the second line is also perpendicular to the first. For this reason, we may speak of two lines as being perpendicular (to each other) without specifying an order.

Perpendicularity easily extends to segments and rays. For example, a line segment

A

B

¯

{displaystyle {overline {AB}}}

is perpendicular to a line segment

C

D

¯

{displaystyle {overline {CD}}}

if, when each is extended in both directions to form an infinite line, these two resulting lines are perpendicular in the sense above. In symbols,

A

B

¯

C

D

¯

{displaystyle {overline {AB}}perp {overline {CD}}}

means line segment AB is perpendicular to line segment CD. For information regarding the perpendicular symbol see Up tack.

A line is said to be perpendicular to a plane if it is perpendicular to every line in the plane that it intersects. This definition depends on the definition of perpendicularity between lines.

Two planes in space are said to be perpendicular if the dihedral angle at which they meet is a right angle (90 degrees).

Perpendicularity is one particular instance of the more general mathematical concept of orthogonality; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word “perpendicular” is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its normal.

• Orthogonal

In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms. Two elements u and v of a vector space with bilinear form B are orthogonal when B(u, v) = 0. Depending on the bilinear form, the vector space may contain nonzero self-orthogonal vectors. In the case of function spaces, families of orthogonal functions are used to form a basis.

By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in other fields including art and chemistry.

Wikipedia

At or forming a right angle (to something).

“In most houses, the walls are perpendicular to the floor.”

Exactly upright; extending in a straight line toward the centre of the earth, etc.

• Perpendicular (noun)

A line or plane that is perpendicular to another.

• Perpendicular (noun)

A device such as a plumb line that is used in making or marking a perpendicular line.

Of two objects, at right angles; perpendicular to each other.

“A chord and the radius that bisects it are orthogonal.”

Statistically independent, with reference to variates.

Of two or more aspects of a problem, able to be treated separately.

“The content of the message should be orthogonal to the means of its delivery.”

Of two or more problems or subjects, independent of or irrelevant to each other.

Wiktionary

at an angle of 90° to a given line, plane, or surface or to the ground

“dormers and gables that extend perpendicular to the main roofline”

at an angle of 90° to the ground; vertical

“the perpendicular cliff”

so steep as to be almost vertical

“houses seem to cling by blind faith to the perpendicular hillside”

denoting the latest stage of English Gothic church architecture, prevalent from the late 14th to mid 16th centuries and characterized by broad arches, elaborate fan vaulting, and large windows with vertical tracery

“the Perpendicular style”

“the handsome Perpendicular church of St Andrew”

• Perpendicular (noun)

a straight line at an angle of 90° to a given line, plane, or surface

“at each division draw a perpendicular representing the surface line”

• Perpendicular (noun)

perpendicular position or direction

“the wall declines from the perpendicular a little inward”