Pengguna:Dedhert.Jr/Uji halaman 01/22

Dalam matematika, jarak Euklides antara kedua titik di ruang Euklides adalah panjang dari segmen garis antara dua titik, dengan koordinat Cartesius dari titik dapat dihitung menggunakan teorema Pythagoras. Jarak tersebut terkadang disebut jarak Pythagoras, yang dinamai dari matematikawan asal Yunani kuno yang bernama Euklides dan Pythagoras. Namun sayangnya, Euklides tidak merepresentasikan jarak sebagai bilangan serta hubungan mengenai perhitungan jarak menggunakan teorema Pythagoras belum ditemukan hingga pada abad ke-18.

Secara umum, jarak antara dua objek yang bukan berupa titik didefinisikan sebagai jarak terkecil di antara pasangan dari titik dadri dua objek. Rumus-rumusnya dikenal sebagai menghitung jarak antara objek yang berbeda, di antaranya: jarak dari titik ke garis. Konsep jarak dalam matematika tingkat lanjut diperumum ke ruang metrik yang abstrak, dan jarak selain Euklides telah dipelajari. Dalam beberapa penerapan seperti statistika dan optimisasi, the square of the Euclidean distance is used instead of the distance itself.

Rumus

Rumus jarak dimensi satu

Jarak antara dua titik pada garis real merupakan nilai mutlak dari selisih numerik koordinat. Dengan kata lain, jika $p$ dan $q$ adalah dua titik pada garis real, maka jarak antara kedua titik tersebut dirumuskan sebagai:^[1] $d(p,q)=|p-q|.$ Adapun rumus yang lebih rumit dengan nilai yang sama seperti sebelumnya, tetapi dapat diperumum ke dimensi yang lebih tinggi dengan mudah:^[1] $d(p,q)={\sqrt {(p-q)^{2}}}.$ Pada rumus di atas, menguadratkan serta membatalkan akar kuadrat memberikan setiap bilangan positif yang tidak diubah, tetapi menggantikan setiap bilangan negatif melalui nilai mutlaknya.^[1]

Rumus jarak dimensi dua

Misalkan titik $p$ mempunyai koordinat Cartesius $(p 1, p 2)$ dan misalkan $q$ mempunyai koordinat $(q 1, q 2)$ di bidang Euklides. Jarak antara $p$ dan $q$ dinyatakan dengan:^[2] $d(p,q)={\sqrt {(q_{1}-p_{1})^{2}+(q_{2}-p_{2})^{2}}}.$ Rumus ini dapat diperlihatkan dengan menerapkan teorema Pythagoras ke segitiga siku-siku yang mempunyai sisi horizontal dan vertikal, serta segmen garis dari $p$ dan $q$ sebagai hipotenusanya. The two squared formulas inside the square root give the areas of squares on the horizontal and vertical sides, and the outer square root converts the area of the square on the hypotenuse into the length of the hypotenuse.^[3]

It is also possible to compute the distance for points given by polar coordinates. If the polar coordinates of $p$ are $(r,\theta )$ and the polar coordinates of $q$ are $(s,\psi )$ , then their distance is^[2] given by the law of cosines: $d(p,q)={\sqrt {r^{2}+s^{2}-2rs\cos(\theta -\psi )}}.$

When $p$ and $q$ are expressed as complex numbers in the complex plane, the same formula for one-dimensional points expressed as real numbers can be used, although here the absolute value sign indicates the complex norm:^[4] $d(p,q)=|p-q|.$

Higher dimensions

Deriving the

n

-dimensional Euclidean distance formula by repeatedly applying the Pythagorean theorem

In three dimensions, for points given by their Cartesian coordinates, the distance is $d(p,q)={\sqrt {(p_{1}-q_{1})^{2}+(p_{2}-q_{2})^{2}+(p_{3}-q_{3})^{2}}}.$ In general, for points given by Cartesian coordinates in $n$ -dimensional Euclidean space, the distance is^[5]

$d(p,q)={\sqrt {(p_{1}-q_{1})^{2}+(p_{2}-q_{2})^{2}+\cdots +(p_{i}-q_{i})^{2}+\cdots +(p_{n}-q_{n})^{2}}}.$

Objects other than points

For pairs of objects that are not both points, the distance can most simply be defined as the smallest distance between any two points from the two objects, although more complicated generalizations from points to sets such as Hausdorff distance are also commonly used.^[6] Formulas for computing distances between different types of objects include:

The distance from a point to a line, in the Euclidean plane^[7]
The distance from a point to a plane in three-dimensional Euclidean space^[7]
The distance between two lines in three-dimensional Euclidean space^[8]

^ ^a ^b ^c Smith, Karl (2013), Precalculus: A Functional Approach to Graphing and Problem Solving, Jones & Bartlett Publishers, hlm. 8, ISBN 978-0-7637-5177-7
^ ^a ^b Cohen, David (2004), Precalculus: A Problems-Oriented Approach (edisi ke-6th), Cengage Learning, hlm. 698, ISBN 978-0-534-40212-9
^ Aufmann, Richard N.; Barker, Vernon C.; Nation, Richard D. (2007), College Trigonometry (edisi ke-6th), Cengage Learning, hlm. 17, ISBN 978-1-111-80864-8
^ Andreescu, Titu; Andrica, Dorin (2014), "3.1.1 The Distance Between Two Points", Complex Numbers from A to ... Z (edisi ke-2nd), Birkhäuser, hlm. 57–58, ISBN 978-0-8176-8415-0
^ Tabak, John (2014), Geometry: The Language of Space and Form, Facts on File math library, Infobase Publishing, hlm. 150, ISBN 978-0-8160-6876-0
^ Ó Searcóid, Mícheál (2006), "2.7 Distances from Sets to Sets", Metric Spaces, Springer Undergraduate Mathematics Series, Springer, hlm. 29–30, ISBN 978-1-84628-627-8
^ ^a ^b Ballantine, J. P.; Jerbert, A. R. (April 1952), "Distance from a line, or plane, to a point", Classroom notes, American Mathematical Monthly, 59 (4): 242–243, doi:10.2307/2306514, JSTOR 2306514
^ Bell, Robert J. T. (1914), "49. The shortest distance between two lines", An Elementary Treatise on Coordinate Geometry of Three Dimensions (edisi ke-2nd), Macmillan, hlm. 57–61

[smith-1] Smith, Karl (2013), Precalculus: A Functional Approach to Graphing and Problem Solving, Jones & Bartlett Publishers, hlm. 8, ISBN 978-0-7637-5177-7

[cohen-2] Cohen, David (2004), Precalculus: A Problems-Oriented Approach (edisi ke-6th), Cengage Learning, hlm. 698, ISBN 978-0-534-40212-9

[3] Aufmann, Richard N.; Barker, Vernon C.; Nation, Richard D. (2007), College Trigonometry (edisi ke-6th), Cengage Learning, hlm. 17, ISBN 978-1-111-80864-8

[4] Andreescu, Titu; Andrica, Dorin (2014), "3.1.1 The Distance Between Two Points", Complex Numbers from A to ... Z (edisi ke-2nd), Birkhäuser, hlm. 57–58, ISBN 978-0-8176-8415-0

[5] Tabak, John (2014), Geometry: The Language of Space and Form, Facts on File math library, Infobase Publishing, hlm. 150, ISBN 978-0-8160-6876-0

[6] Ó Searcóid, Mícheál (2006), "2.7 Distances from Sets to Sets", Metric Spaces, Springer Undergraduate Mathematics Series, Springer, hlm. 29–30, ISBN 978-1-84628-627-8

[baljer-7] Ballantine, J. P.; Jerbert, A. R. (April 1952), "Distance from a line, or plane, to a point", Classroom notes, American Mathematical Monthly, 59 (4): 242–243, doi:10.2307/2306514, JSTOR 2306514

[8] Bell, Robert J. T. (1914), "49. The shortest distance between two lines", An Elementary Treatise on Coordinate Geometry of Three Dimensions (edisi ke-2nd), Macmillan, hlm. 57–61

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