Distance - mile

Japanese: 距離 - きょり

In mathematics, distance is defined in various cases, as follows: (1) Distance between two points The shortest line connecting two points A and B is the line segment AB, and its length is the distance between two points A and B. (2) Distance between a point and a line The shortest line connecting point A and any point on line l is the perpendicular line AH drawn from A to l, and its length is called the distance between A and l. (3) Distance between a point and a plane The shortest line connecting point A and any point on plane α is the perpendicular line AH drawn from A to α, and its length is called the distance between A and α. (4) Distance between two parallel lines For two parallel lines l and m, if the points A and B where a line perpendicular to both intersects with l and m are A and B, then the length of the line segment AB is constant wherever the common perpendicular line is taken. This length is called the distance between the two parallel lines l and m. (5) Distance between two parallel planes If the points A and B where a line perpendicular to both planes α and β intersect are A and B, then the length of the line segment AB is called the distance between the two planes α and β. (6) Distance between two lines in a twisted position If the two lines are p and q, and an arbitrary point P is taken on p, and an arbitrary point Q is taken on q, then the shortest line connecting P and Q is the line segment PQ that is perpendicular to both p and q, and this is called the distance between the two lines p and q. In this case, there is only one line PQ that is perpendicular to both p and q, and it is called the common perpendicular line of p and q.

[Minoru Kurita]

Coordinates and distances

(1) On a plane, when the Cartesian coordinates of two points A and B are (a ₁ , a ₂ ) and (b ₁ , b ₂ ), the distance is,

In space, if the Cartesian coordinates of two points A and B are (a ₁ , a ₂ , a ₃ ) and (b ₁ , b ₂ , b ₃ ), then the distance is

(2) In Cartesian coordinates, if the coordinates of point P are ( _x1 , _y1 ) and the equation of line p is ax+by+c=0, then the distance PH between P and p is,

In Cartesian coordinates, if the coordinates of point P are (x ₁ , y ₁ , z ₁ ) and the equation of plane α is ax + by + cz + d = 0, then the distance PH between P and α is:

It is.

[Minoru Kurita]

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

数学では、以下のようないろいろな場合に分けて距離を定義する。(1)2点間の距離　2点A、Bを結ぶ線のなかでもっとも短いのは線分ABで、その長さが2点A、B間の距離である。(2)点と直線の距離　点Aと直線l上の任意の点とを結ぶ線のなかでもっとも短いのはAからlへ下ろした垂線AHで、その長さをAとlの距離という。(3)点と平面の距離　点Aと平面α上の任意の点とを結ぶ線のなかでもっとも短いのは、Aからαへ下ろした垂線AHで、その長さをAとαの距離という。(4)平行2直線の距離　平行な2直線l、mについて、両方に垂直な直線がl、mと交わる点をA、Bとすると、線分ABの長さは共通の垂線をどこにとっても一定である。この長さを平行2直線l、mの距離という。(5)平行2平面の距離　平行な2平面α、βについて、両方に垂直な直線がα、βと交わる点をA、Bとするとき、線分ABの長さを2平面α、βの距離という。(6)ねじれの位置にある2直線の距離　2直線をp、qとし、p上に任意の点P、q上に任意の点Qをとるとき、P、Qを結ぶ線のなかでもっとも短いのは、p、qの両方に垂直な線分PQで、これを2直線p、qの距離という。このときのp、qの両方に直交する直線PQはただ一つあって、p、qの共通垂線という。

［栗田　稔］

座標と距離

(1)平面上で、2点A、Bの直交座標が（a₁, a₂), (b₁, b₂）のとき、距離は、

空間で、2点A、Bの直交座標が(a₁, a₂, a₃), (b₁, b₂, b₃)のとき、距離は、

(2)直交座標で、点Pの座標が（x₁, y₁）、直線pの方程式がax＋by＋c＝0のとき、Pとpの距離PHは、

直交座標で、点Pの座標が（x₁, y₁, z₁）、平面αの方程式がax＋by＋cz＋d＝0のとき、Pとαの距離PHは、

である。