Hubbry Logo
Right angleRight angleMain
Open search
Right angle
Community hub
Right angle
logo
8 pages, 0 posts
0 subscribers
Be the first to start a discussion here.
Be the first to start a discussion here.
Right angle
Right angle
Right angle
View on Wikipedia
from Wikipedia
For other uses, see Right angle (disambiguation).
A right angle is equal to 90 degrees.
A line segment (AB) drawn so that it forms right angles with a line (CD)
Types of angles
2D angles

Right
Interior
Exterior

Spherical
2D angle pairs

Adjacent
Vertical
Complementary
Supplementary

Transversal
3D angles

Dihedral

Solid

In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or ⁠/2⁠ radians[1] corresponding to a quarter turn.[2] If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles.[3] The term is a calque of Latin angulus rectus; here rectus means "upright", referring to the vertical perpendicular to a horizontal base line.

Closely related and important geometrical concepts are perpendicular lines, meaning lines that form right angles at their point of intersection, and orthogonality, which is the property of forming right angles, usually applied to vectors. The presence of a right angle in a triangle is the defining factor for right triangles,[4] making the right angle basic to trigonometry.

Etymology

[edit]

The meaning of right in right angle possibly refers to the Latin adjective rectus 'erect, straight, upright, perpendicular'. A Greek equivalent is orthos 'straight; perpendicular' (see orthogonality).

In elementary geometry

[edit]

A rectangle is a quadrilateral with four right angles. A square has four right angles, in addition to equal-length sides.

The Pythagorean theorem states how to determine when a triangle is a right triangle.

Symbols

[edit]
Right triangle, with the right angle shown via a small square
Another option of diagrammatically indicating a right angle, using an angle curve and a small dot

In Unicode, the symbol for a right angle is U+221F ∟ RIGHT ANGLE (∟). It should not be confused with the similarly shaped symbol U+231E ⌞ BOTTOM LEFT CORNER (⌞, ⌞). Related symbols are U+22BE ⊾ RIGHT ANGLE WITH ARC (⊾), U+299C ⦜ RIGHT ANGLE VARIANT WITH SQUARE (⦜), and U+299D ⦝ MEASURED RIGHT ANGLE WITH DOT (⦝).[5]

In diagrams, the fact that an angle is a right angle is usually expressed by adding a small right angle that forms a square with the angle in the diagram, as seen in the diagram of a right triangle (in British English, a right-angled triangle) to the right. The symbol for a measured angle, an arc, with a dot, is used in some European countries, including German-speaking countries and Poland, as an alternative symbol for a right angle.[6]

Euclid

[edit]

Right angles are fundamental in Euclid's Elements. They are defined in Book 1, definition 10, which also defines perpendicular lines. Definition 10 does not use numerical degree measurements but rather touches at the very heart of what a right angle is, namely two straight lines intersecting to form two equal and adjacent angles.[7] The straight lines which form right angles are called perpendicular.[8] Euclid uses right angles in definitions 11 and 12 to define acute angles (those smaller than a right angle) and obtuse angles (those greater than a right angle).[9] Two angles are called complementary if their sum is a right angle.[10]

Book 1 Postulate 4 states that all right angles are equal, which allows Euclid to use a right angle as a unit to measure other angles with. Euclid's commentator Proclus gave a proof of this postulate using the previous postulates, but it may be argued that this proof makes use of some hidden assumptions. Saccheri gave a proof as well but using a more explicit assumption. In Hilbert's axiomatization of geometry this statement is given as a theorem, but only after much groundwork. One may argue that, even if postulate 4 can be proven from the preceding ones, in the order that Euclid presents his material it is necessary to include it since without it postulate 5, which uses the right angle as a unit of measure, makes no sense.[11]

Conversion to other units

[edit]

A right angle may be expressed in different units:

Rule of 3-4-5

[edit]

Throughout history, carpenters and masons have known a quick way to confirm if an angle is a true right angle. It is based on the Pythagorean triple (3, 4, 5) and the rule of 3-4-5. From the angle in question, running a straight line along one side exactly three units in length, and along the second side exactly four units in length, will create a hypotenuse (the longer line opposite the right angle that connects the two measured endpoints) of exactly five units in length.

Thales' theorem

[edit]
Construction of the perpendicular to the half-line h from the point P (applicable not only at the end point A, M is freely selectable), animation at the end with pause 10 s
Alternative construction if P outside of the half-line h and the distance A to P' is small (B is freely selectable),
animation at the end with pause 10 s
Main article: Thales' theorem

Thales' theorem states that an angle inscribed in a semicircle (with a vertex on the semicircle and its defining rays going through the endpoints of the semicircle) is a right angle.

Two application examples in which the right angle and the Thales' theorem are included (see animations).

Generalizations

[edit]

The solid angle subtended by an octant of a sphere (the spherical triangle with three right angles) equals π/2  sr.[12]

See also

[edit]
Wikimedia Commons has media related to Right angles.

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Right angle

View on Grokipedia
from Grokipedia
A right angle is an angle formed by the intersection of two rays, line segments, or lines that measures exactly 90 degrees (or π/2\pi/2 radians). In diagrams, it is conventionally marked by a small square at the vertex to indicate its precise 90-degree measure. The concept of the right angle originates in , where formalized it in his Elements (circa 300 BCE) through Postulate 4, stating that all right angles are equal to one another, establishing them as a universal standard for angular equality without needing measurement tools. This postulate underpins ity: two lines are perpendicular if they form right angles, a property that ensures the adjacent angles created by their intersection are congruent (each 90 degrees) and pairs of adjacent angles sum to 180 degrees. In , right angles define key structures such as right triangles, where one angle is 90 degrees and are acute, summing to 90 degrees; this configuration enables the , relating the sides as a2+b2=c2a^2 + b^2 = c^2 (with cc the ). Right angles also appear in coordinate systems, where the x- and y-axes intersect at 90 degrees, facilitating Cartesian geometry and vector analysis. Beyond , right angles are essential in and applied sciences for ensuring , such as in the design of supports in bridges, , and electrical circuits, where deviations can compromise load distribution. They form the basis of in right triangles, enabling calculations for heights, distances, and forces in fields like , , and physics.

Definition and Fundamentals

Etymology

The term "right angle" originates from the Latin phrase angulus rectus, where rectus denotes "upright" or "straight," and refers to "angle" or "corner." This phrasing emphasized the perpendicular alignment inherent in the , translating directly into various European languages as a . Ancient terminology influenced early geometric discourse, using orthē gōnia (ὀρθὴ γωνία), derived from orthos meaning "straight," "upright," or "right," combined with gōnia for "angle." This appears in related terms like orthogōnios (right-angled), underscoring the idea of proper or vertical straightness in classical texts. In English, the compound "right angle" first appeared in the late 14th century during the period, evolving from "ryght angle" around 1400, where "right" (from riht, meaning straight or proper, rooted in Proto-Indo-European *reg- "to move in a straight line") paired with "angle" (borrowed from angle, itself from Latin angulus). This usage reflected emerging translations of geometric ideas tied to perpendicularity. Related terms persist in other languages, such as French angle droit (with droit from Latin directus, "straight") and German Rechtwinkel (from recht, meaning straight or right, and Winkel, angle).

Basic Properties in Geometry

A right angle is defined as an angle formed by two rays, line segments, or lines that share a common endpoint, known as the vertex, and measure exactly 90∘90^\circ or π/2\pi/2 radians. This measurement corresponds to a quarter turn, equivalent to one-fourth of a full circle. One of the fundamental properties of a right angle is that it is formed by two lines, which such that they create four congruent right angles at the point of . When two such lines extend infinitely in a plane, they divide the plane into four equal regions called quadrants. In the context of triangles, a right angle at one vertex defines a right triangle, where the side opposite the right angle is the hypotenuse, and the other two sides are the legs. Right angles relate to other angle types through addition: two complementary angles, each less than 90∘90^\circ, sum to exactly 90∘90^\circ to form a right angle. Additionally, two right angles are supplementary, summing to 180∘180^\circ. In basic polygonal shapes, rectangles and squares each possess four right angles, with squares also having equal sides. Furthermore, in coordinate geometry, the standard Cartesian axes are perpendicular, intersecting at the origin to form right angles that establish the framework for plotting points in the plane.

Representation and Notation

Symbols and Diagramming

In , the right angle is commonly represented by the Unicode character ∟ (U+221F), which belongs to the Mathematical Operators block and depicts an L-shaped indicating a 90-degree angle. This is widely used in and to denote a quarter turn or perpendicular intersection. Alternatively, the general angle ∠ (U+2220) may be adapted for right angles by specifying "90°" or a similar qualifier, though the dedicated ∟ is preferred for precision in formal texts. Diagramming conventions for right angles emphasize visual clarity at the vertex where two lines meet . The most standard method involves placing a small square mark at the intersection point, symbolizing the perfect fit of a square's corner and confirming the 90-degree measure without numerical labels. In some illustrations, an L-shaped mark mirroring the ∟ is used directly, while less common variants include an arc across the angle with a central dot to highlight the right angle's position. These conventions ensure that lines forming the right angle are immediately recognizable in sketches and proofs. In historical proofs and illustrations, right angles were denoted descriptively rather than symbolically in ancient texts like Euclid's Elements, where they are defined as angles formed by a line standing on another such that adjacent angles are equal, accompanied by simple line diagrams labeling points without modern marks. Modern textbooks retain this diagrammatic tradition but incorporate the small square or ∟ for emphasis, evolving from hand-drawn figures—often etched on wax tablets or —to printed engravings in the and now interactive digital representations in software like . This progression allows for scalable, annotated visuals in educational and research contexts. For typing and software notation, the ∟ symbol can be inserted via keyboard shortcuts in Unicode-enabled environments, such as Alt+8735 on Windows, or through markup languages. In , the command \rightangle from the stix package renders the symbol, as in \rightangle\rightangle, facilitating its inclusion in mathematical documents. Tools like TikZ in further enable custom diagramming of right angles with the small square mark using coordinates and paths.

Measurement Units

The right angle is quantified as 90 degrees in the system, a base-60 measurement originating from ancient around the second millennium BCE, where the full circle was divided into 360 parts based on the approximate 360-day solar year. This subdivision allowed for precise tracking of celestial movements, with each degree further divided into and each minute into . In this system, the right angle represents exactly one-quarter of a full . In radian measure, the right angle equals π/2\pi/2 , a unit defined by the ratio of to on a , providing a dimensionally consistent approach preferred in advanced and . The conversion between degrees and radians follows the formula θrad=θdeg×π180\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}, so for the right angle, 90∘=π2≈1.570890^\circ = \frac{\pi}{2} \approx 1.5708 . This equivalence stems from the full circle measuring 2π2\pi or 360 degrees. Alternative units include gradians (or gons), where the right angle measures 100 gradians, as a full circle comprises 400 gradians in the centesimal system designed for decimal compatibility in and . In terms of turns, it is 14\frac{1}{4} turn, reflecting a simple fractional division of the complete rotation. The military mil (short for ) uses 1600 mils for the right angle, with 6400 mils in a full circle, approximating the for and targeting precision. In trigonometric calculations, the right angle's measure ensures precise values such as sin⁡(90∘)=1\sin(90^\circ) = 1 and cos⁡(90∘)=0\cos(90^\circ) = 0, foundational identities derived from the unit circle where the is the y-coordinate and cosine the x-coordinate at that . These properties underpin computations in fields requiring angular precision, emphasizing the importance of consistent unit application to avoid errors in conversions.

Euclidean Geometry Context

Elementary Applications

In basic geometry, right angles form the foundation of common polygonal shapes. A is defined as a with four right angles and opposite sides of equal length. This property ensures that the figure has parallel opposite sides and equal diagonals, making it a fundamental shape in planar . A represents a special case of the , where all four sides are of equal length, resulting in all angles being right angles. Squares exhibit additional symmetries, such as diagonals that are equal in length and to each other, which arise directly from the uniform right angles at each vertex. Right triangles provide another elementary application, characterized by exactly one right angle, with the side opposite this angle known as the hypotenuse. The relationship between the sides aa, bb (the legs), and cc (the hypotenuse) is given by the Pythagorean theorem: a2+b2=c2a^2 + b^2 = c^2 This equation allows for the calculation of unknown side lengths in right triangles, forming the basis for many introductory problems in distance and measurement. The relies on right angles for its structure, where the x-axis and y-axis intersect perpendicularly at the origin to define positions in a plane. This orthogonal arrangement enables the graphing of points, lines, and curves by measuring distances along these axes, facilitating applications in plotting functions and analyzing spatial relationships in . Simple geometric constructions also demonstrate the utility of right angles using only a and . To draw a line through a point P on a given line, mark two points A and B on the line equidistant from P using a centered at P with arbitrary . Then, with the compass set to AB, draw arcs centered at A and at B; these arcs intersect at a point Q off the line. The line PQ forms a right angle with the given line. This method produces lines that intersect at 90 degrees, essential for creating accurate diagrams and verifying perpendicularity in basic exercises.

Euclid's Formalization

In Euclid's Elements, composed around 300 BCE, the right angle is formally defined in Book I, Definition 10 as follows: "When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right." This definition establishes the right angle through the equality of adjacent angles formed by a transversal line intersecting another, emphasizing a relational property rather than an absolute measure, which aligns with the axiomatic approach avoiding numerical quantification. By framing the right angle in terms of congruence between supplementary angles, Euclid provides a foundational concept that underpins subsequent geometric constructions without relying on external measurement tools. Complementing this definition, Postulate 4 asserts that "all right angles are equal to one another," a principle that guarantees the congruence of right angles across any context in plane geometry. This postulate is crucial because it dispenses with the need to measure or compare right angles individually, allowing to treat them as inherently equivalent in proofs and constructions. It establishes a uniformity that permeates the Elements, enabling the derivation of more complex properties from simpler assumptions and reinforcing the system's reliance on logical deduction over empirical verification. Right angles feature prominently in early propositions of Book I, illustrating their practical role within the axiomatic framework. Proposition 11 demonstrates how to erect a at a given point on a straight line, constructing a right angle through a series of formations and line extensions, thereby providing a method to generate right angles on demand. Propositions 31 and 32 further integrate right angles into the study of parallels and triangles: Proposition 31 shows that a line through a point parallel to a given line can be drawn by constructing it such that the alternate interior angles formed with a transversal are equal to each other, while Proposition 32 proves that the interior angles of any triangle sum to two right angles, linking right angles to the global structure of polygonal figures. These propositions highlight the right angle's centrality as a building block for parallelism and angular relations, essential for advancing . Euclid's formalization of the right angle profoundly shaped Western , serving as a cornerstone of for over two millennia and influencing fields from to astronomy. The Elements became the standard text for geometric education in and the Islamic world, with its treatment of right angles exemplifying rigorous proof that inspired later developments, including and the foundations of . This axiomatic approach to right angles ensured their enduring role in mathematical pedagogy and theoretical frameworks until the emergence of non-Euclidean geometries in the 19th century.

Theorems and Constructions

Thales' Theorem

Thales' theorem states that if A and B are the endpoints of a of a , and C is any point on the circumference of the , then the angle ∠ACB is a right angle. This result is attributed to (c. 624–546 BCE), considered one of the earliest recorded theorems in Greek geometry, based on accounts from ancient sources such as citing Pamphile. The proof relies on properties of isosceles triangles and the geometry of the circle. Let O be the center of the circle, so OA = OB = OC as radii. Triangles AOC and BOC are isosceles with OA = OC and OB = OC, respectively. Let the base angles at A and C in triangle AOC be equal, denoted α, so angle OAC = angle OCA = α. Similarly, let the base angles at B and C in triangle BOC be equal, denoted β, so angle OBC = angle OCB = β. Since A, O, B are collinear along the diameter, ∠AOC + ∠COB = 180°. The sum of angles in triangle AOC gives ∠AOC + 2α = 180°. Similarly, ∠COB + 2β = 180°. Adding these equations: (∠AOC + ∠COB) + 2(α + β) = 360°, so 180° + 2(α + β) = 360°, thus α + β = 90°. The angle ∠ACB is the sum of angles OCA and OCB, which is α + β = 90°. This enables the of right triangles given a as the : draw a with the segment as diameter and select any point C on the to form ∠ACB = 90° using and . The converse also holds: if a triangle has a right angle at C, then the hypotenuse AB serves as the diameter of the unique circle circumscribing the .

3-4-5 Rule

The 3-4-5 rule utilizes a consisting of sides measuring 3 units, 4 units, and 5 units, where the relationship 32+42=9+16=25=523^2 + 4^2 = 9 + 16 = 25 = 5^2 holds, verifying a between the 3-unit and 4-unit sides. This simple proportion allows for the empirical confirmation or creation of a 90-degree using basic measuring tools, without requiring protractors or advanced instruments. Ancient and Babylonians employed this method in and by using ropes knotted at intervals corresponding to 3, 4, and 5 units, enabling the layout of right angles for buildings and fields. For instance, Egyptian evidence appears in the Rhind Papyrus (c. 1650 BCE), which includes problems solvable via 3-4-5 proportions, while Babylonian tablets like (c. 1800 BCE) list related triples for practical geometric tasks. These cultures integrated the technique into "squaring" processes to ensure perpendicular alignments in and land measurement. To apply the 3-4-5 rule for constructing a , start at a reference point and measure 3 units along one line; from the same point, measure 4 units along the proposed line; then, measure the diagonal distance between the endpoints of these segments and adjust until it equals exactly 5 units, at which point the corner is a verified right angle. This iterative process relies on taut ropes or tapes for marking and can be performed with stakes or chalk lines on-site. The rule's scalability stems from its basis in similar triangles, so any multiple of the 3-4-5 proportions—such as 6-8-10 or 9-12-15—preserves the right angle property, as the ratios remain identical. In real-world measurements, larger multiples enhance error tolerance by amplifying absolute lengths, which reduces the proportional impact of minor inaccuracies in tools like tapes or ropes, making the method more reliable for bigger structures.

Practical and Modern Applications

Conversions and Calculations

In right triangle trigonometry, the right angle at 90° serves as a reference for defining key ratios, where the sine of 90° equals 1, the cosine equals 0, and the is undefined due to in the adjacent-over-opposite formulation. These values arise from the , where at 90°, the vertical coordinate is 1 and the horizontal is 0, making the right angle pivotal for computing opposite, adjacent, and sides in applications like height estimation. Practical conversions of right angles extend to everyday systems, such as timekeeping on analog clocks, where a full circle of 360° divides into 12 hours, yielding 30° per hour; thus, a 90° angle corresponds to a 3-hour separation between hour markers. In , a 90° turn represents a quarter-circle change in direction, often used for course corrections, as in land or nautical bearings where east from north is exactly 90°. A right angle also equates to π/2 radians, facilitating seamless unit shifts in computational models. Surveying calculations frequently employ right angles to establish baselines, using tools like tapes and poles to offset measurements at 90° from a primary line, ensuring accurate grid layouts for mapping. propagation in these angle measurements follows differential rules, where small angular deviations δθ amplify positional errors as d increases, roughly by δx ≈ d · δθ (in radians), necessitating precise to minimize traverse misclosures. In software like CAD systems, 90° inputs trigger polar tracking alignments, automatically snapping lines and objects to perpendicular orientations for efficient drafting of orthogonal structures. Calculators handle these via built-in trig functions, instantly yielding ratios for 90° without manual computation, supporting rapid iterations in design workflows.

Uses in Engineering and Architecture

In architecture, right angles are fundamental for ensuring by creating plumb walls and right-angled corners that distribute loads evenly and resist lateral forces such as or earthquakes. These perpendicular alignments allow buildings to maintain vertical , preventing leaning or collapse over time, as seen in the orthogonal frameworks of modern and residential structures. For staircases, the 90-degree angle between risers and treads provides a stable, predictable footing that enhances user by minimizing trip hazards and ensuring uniform step . Occupational safety standards, such as OSHA 1910.25, specify maximum riser heights of 9.5 inches (24 cm) and minimum tread depths of 9.5 inches (24 cm) for standard . In , perpendicular beams in bridge supports facilitate efficient load transfer and structural rigidity, with cross-frames positioned at right angles to girders to counteract torsional forces and maintain alignment under dynamic loads like . Similarly, in electronics engineering, circuit board layouts often employ orthogonal right-angle traces to optimize space and efficiency, though careful mitigates potential signal reflections at high frequencies to preserve integrity. Construction techniques rely on right angles for precise framing, where tools like spirit levels and carpenter squares verify perpendicularity between studs and plates to achieve square assemblies that support sheathing and finishes without distortion. Diagonal bracing, calculated using principles like the 3-4-5 method derived from the , reinforces these frames by spanning hypotenuses across right-angled corners, enhancing shear resistance in walls and roofs. In modern technologies, right angles enable precise orthogonal movements in , particularly Cartesian systems that traverse X, Y, and axes at 90 degrees for accurate pick-and-place operations in . GPS navigation systems frequently incorporate 90-degree turn instructions in urban grid-based routing, allowing vehicles to follow efficient paths while accounting for real-time orientation adjustments.

Advanced Generalizations

In Higher Dimensions

In three-dimensional , the concept of a right angle extends beyond lines to include , which are the angles formed between two intersecting planes measuring exactly 90 degrees. A of 90 degrees occurs when the normal vectors to the two planes are perpendicular, satisfying the condition where the cosine of the angle θ is zero, as derived from the of the normals. This generalization allows for the description of perpendicularity in , such as the faces of polyhedra where adjacent planes meet orthogonally. Another extension in three dimensions involves solid angles, which measure the angular extent of a three-dimensional region from a point. The octant of a , bounded by three mutually planes meeting at the origin, subtends a of π/2 steradians, equivalent to one-eighth of the full 's 4π steradians. This right-angled is fundamental in contexts like radiation patterns or coordinate divisions in 3D space. In Cartesian coordinate systems, orthogonality in three dimensions is formalized using the vectors along the x-, y-, and z-axes, which are mutually . Two vectors u\mathbf{u} and v\mathbf{v} in R3\mathbb{R}^3 are orthogonal if their is zero: u⋅v=u1v1+u2v2+u3v3=0\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 = 0, confirming they form right angles in the Euclidean metric. This property underpins the structure of 3D orthogonal coordinate systems, enabling precise spatial decompositions. The notion of right angles generalizes further to n-dimensional Euclidean spaces through the framework of linear algebra, where is defined pairwise via the inner product. In Rn\mathbb{R}^n, an consists of n vectors {e1,…,en}\{\mathbf{e}_1, \dots, \mathbf{e}_n\} such that ei⋅ej=0\mathbf{e}_i \cdot \mathbf{e}_j = 0 for i≠ji \neq j, with each pair forming a right angle, and the vectors are typically normalized to unit length for an . This construction is essential for representing spaces with mutually perpendicular directions, facilitating computations in high-dimensional and . A representative example in three dimensions is the , where the 12 edges meet at vertices such that adjacent faces form dihedral angles of exactly 90 degrees, embodying perfect in solid form. In higher dimensions, this generalizes to the or n-cube, an orthotope with facets that are (n-1)- meeting at right dihedral angles along edges, preserving perpendicularity across all dimensions; for instance, the 4-dimensional features 32 edges and 24 square faces aligned orthogonally. These structures illustrate how right angles scale to higher-dimensional polytopes, maintaining the Euclidean perpendicularity principle.

In Non-Euclidean and Vector Spaces

In non-Euclidean geometries, the concept of a right angle adapts to curved spaces while retaining its foundational role in defining perpendicularity. In , right angles are defined axiomatically as congruent to one another, consistent with the parallel postulate that permits multiple lines through a point parallel to a given line, leading to angle sums in triangles less than 180 degrees. This contrasts with but preserves the local notion of perpendicularity between geodesics. For instance, hyperbolic right triangles satisfy modified trigonometric identities where the behaves differently due to the negative . Spherical geometry, the geometry of the sphere's surface, defines right angles between great circles—the shortest paths analogous to straight lines—when their defining planes through the sphere's are . A classic example is the of the and a meridian, forming a 90-degree angle at their points on the , as measured by the between the planes. In , right-angled triangles have angle sums exceeding 180 degrees, with applications in and astronomy where great-circle routes intersect orthogonally. In abstract vector spaces equipped with an inner product, generalizes the right angle: two vectors u\mathbf{u} and v\mathbf{v} are orthogonal if their inner product satisfies ⟨u,v⟩=0\langle \mathbf{u}, \mathbf{v} \rangle = 0, preserving the geometric intuition of perpendicularity without requiring a Euclidean metric. This framework extends to Hilbert spaces in , where orthogonal bases decompose functions into non-overlapping components. In , orthogonal states ∣ψ⟩|\psi\rangle and ∣ϕ⟩|\phi\rangle exhibit zero overlap via ⟨ψ∣ϕ⟩=0\langle \psi | \phi \rangle = 0, enabling distinguishable measurements and forming the basis for and entanglement analyses. In the context of , Minkowski space introduces a pseudo-Euclidean metric where right angles represent perpendicularity between spacelike directions, visualized in diagrams as 45-degree light cones bounding causal structures. here uses the Minkowski inner product ημνuμvν=0\eta_{\mu\nu} u^\mu v^\nu = 0 for two four-vectors uu and vv, essential for transforming coordinates without altering physical invariants. The (GPS) incorporates corrections for these non-Euclidean effects from general relativity's curved , including and velocity-based shifts, achieving sub-meter accuracy by adjusting satellite clock rates by about 38 microseconds daily. Modern applications in leverage orthogonal matrices—square matrices QQ satisfying QTQ=IQ^T Q = I—to perform transformations that preserve angles and norms, mitigating issues like vanishing gradients in neural networks. These matrices optimize over Stiefel manifolds for tasks such as and low-rank approximations, ensuring stable training by maintaining geometric structure in high-dimensional data spaces.

References

  1. https://mcckc.edu/tutoring/docs/blue-river/math/basic/Basic_Geometric_Terms.pdf
  2. https://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut33_geom.htm
  3. http://aleph0.clarku.edu/~djoyce/elements/bookI/post4.html
  4. https://www.math.purdue.edu/~goldberg/Math460/Theorem_list_v5.pdf
  5. https://mathresearch.utsa.edu/wiki/index.php?title=Theorem_of_Right_Triangles
  6. https://www.palmbeachstate.edu/prepmathlw/Documents/the_pythagorean_theorem.pdf
  7. https://www.rit.edu/academicsuccesscenter/sites/rit.edu.academicsuccesscenter/files/documents/math-handouts/T1_RightAngleTrig_BP_8_9_16.pdf
  8. https://engineering.usu.edu/students/engineering-math-resource-center/topics/pre-calculus/trigonometry/trigonometry-review
  9. https://math.ou.edu/~alutz/unit-1-right-triangle-trigonometry.pdf
  10. https://www.etymonline.com/word/angle
  11. https://www.etymonline.com/word/orthogonal
  12. https://www.etymonline.com/word/right
  13. https://www.etymonline.com/word/droit
  14. https://mathworld.wolfram.com/RightAngle.html
  15. https://www.cuemath.com/geometry/right-angle/
  16. https://study.com/academy/lesson/what-is-a-right-angle-definition-formula-quiz.html
  17. https://www.elac.edu/sites/elac.edu/files/2022-08/Lines-and-Angles.pdf
  18. https://science.howstuffworks.com/math-concepts/coordinate-plane.htm
  19. https://mathworld.wolfram.com/RightTriangle.html
  20. https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-geometry/cc-7th-angles/a/complementary-and-supplementary-angles-review
  21. https://www.khanacademy.org/math/geometry-home/quadrilaterals-and-polygons/geometry-quads/a/quadrilaterals-review
  22. https://people.computing.clemson.edu/~dhouse/courses/401/notes/coord-systems.pdf
  23. https://www.fileformat.info/info/unicode/char/221f/index.htm
  24. https://mathbitsnotebook.com/Geometry/BasicTerms/BTnotation2.html
  25. https://www.khanacademy.org/math/cc-third-grade-math/quadrilaterals-3rd/imp-quadrilaterals/a/right-angles-in-shapes
  26. http://aleph0.clarku.edu/~djoyce/elements/bookI/bookI.html
  27. https://digitalcommons.chapman.edu/cgi/viewcontent.cgi?article=1019&context=mpp_published_research
  28. https://aneescraftsmanship.com/angle-symbols%25E2%2588%25A0%25E2%2588%259F%25E2%2588%25A1%25E2%2588%25A2-in-latex/
  29. https://tikz.net/right_angle/
  30. https://digitalcommons.ursinus.edu/cgi/viewcontent.cgi?article=1006&context=triumphs_precalc
  31. https://louis.pressbooks.pub/trigonometry/chapter/introduction-to-trig-radians/
  32. https://www.itl.nist.gov/div898/software/dataplot/refman1/ch5/angle_un.pdf
  33. https://ximera.osu.edu/m4t/elementaryTeachersTwo/elementaryReading/Preliminaries/PointsLinesAngles
  34. https://pubs.nctm.org/view/journals/mt/34/5/article-p211.xml
  35. https://www.slcc.edu/math/docs/oer-trigonometry.pdf
  36. https://mathworld.wolfram.com/Rectangle.html
  37. https://mathworld.wolfram.com/Square.html
  38. https://mathworld.wolfram.com/PythagoreanTheorem.html
  39. http://aleph0.clarku.edu/~djoyce/elements/bookI/propI11.html
  40. http://aleph0.clarku.edu/~djoyce/elements/bookI/defI10.html
  41. http://aleph0.clarku.edu/~djoyce/elements/bookI/propI31.html
  42. http://aleph0.clarku.edu/~djoyce/elements/bookI/propI32.html
  43. https://mathshistory.st-andrews.ac.uk/Biographies/Euclid/
  44. https://library.brown.edu/exhibits/archive/math/nofr.html
  45. https://mathshistory.st-andrews.ac.uk/Biographies/Thales/
  46. https://mathcircle.berkeley.edu/sites/default/files/archivedocs/2015/lecture/BMC_IntII_Sep15.pdf
  47. https://sites.pitt.edu/~jdnorton/teaching/paradox/chapters/impossible_geometry/impossible_geometry.html
  48. https://www.mathopenref.com/triangle345.html
  49. https://study.com/academy/lesson/special-right-triangles-3-4-5-triangle.html
  50. http://www.geom.uiuc.edu/~ekeel/Project/hist.html
  51. https://www.dam.brown.edu/people/mumford/BookBlogPosts/Pythagoras%20Rule.pdf
  52. https://people.math.harvard.edu/~knill/teaching/mathe320_2022/handouts/babylon.pdf
  53. https://cld.mgcafe.uky.edu/sites/cld.ca.uky.edu/files/Ag.%20Mechanics%20Shop%20Book.pdf
  54. https://www.storyofmathematics.com/3-4-5-triangle/
  55. https://www.aconcordcarpentry.com/the-3-4-5-method-for-squaring-corners.html
  56. https://tutorial.math.lamar.edu/pdf/trig_cheat_sheet.pdf
  57. https://www.workybooks.com/resources/math-vocabulary/clock-angle-formula
  58. https://www.uakron.edu/armyrotc/ms1/7.pdf
  59. https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:trig/x2ec2f6f830c9fb89:radians/v/we-converting-degrees-to-radians
  60. https://www.fao.org/4/r7021e/r7021e05.htm
  61. https://onlinelibrary.wiley.com/doi/10.1002/9781119390664.ch8
  62. https://www.autodesk.com/learn/ondemand/tutorial/create-angle-measurements-with-polar-tracking
  63. https://www.uniccm.com/blog/angles-in-construction
  64. https://www.98thpercentile.com/blog/understanding-right-angle
  65. https://www.osha.gov/laws-regs/regulations/standardnumber/1910/1910.25
  66. https://www.fhwa.dot.gov/bridge/pubs/hif22034.pdf
  67. https://resources.altium.com/p/pcb-routing-angle-myths-45-degree-angle-versus-90-degree-angle
  68. https://www.homedepot.com/c/ah/how-to-use-levels-and-carpenter-squares/9ba683603be9fa5395fab9031b88896
  69. https://www.johnsonlevel.com/News/FramingSquare
  70. https://www.homebuildingandrepairs.com/math/3_4_5_rule.html
  71. https://standardbots.com/blog/robot-movement-and-motion-types
  72. https://www.backpacker.com/skills/navigation-101-understanding-your-compass/
  73. https://mathworld.wolfram.com/DihedralAngle.html
  74. https://mathworld.wolfram.com/SolidAngle.html
  75. https://textbooks.math.gatech.edu/ila/dot-product.html
  76. https://math.libretexts.org/Bookshelves/Linear_Algebra/A_First_Course_in_Linear_Algebra_(Kuttler)/04%3A_R/4.11%3A_Orthogonality
  77. https://mathworld.wolfram.com/Cube.html
  78. https://mathworld.wolfram.com/Hypercube.html
  79. https://math.osu.edu/sites/math.osu.edu/files/hyperbolicGeometry.pdf
  80. https://documents.kenyon.edu/math/atwell.pdf
  81. https://www.math.ucla.edu/~robjohn/math/spheretrig.pdf
  82. https://www.math.hkust.edu.hk/~mabfchen/Math111/Week13-14.pdf
  83. https://pubs.aip.org/aapt/ajp/article/85/9/683/1038437/The-geometry-of-relativity
  84. https://tf.boulder.nist.gov/general/pdf/1229.pdf
  85. http://proceedings.mlr.press/v32/shalit14.pdf
Add your contribution
Related Hubs
User Avatar
No comments yet.