# Straight line through the origin

In mathematics, a straight line through the origin is a straight line that runs through the coordinate origin of a given Cartesian coordinate system . Therefore straight lines through the origin are described by particularly simple straight line equations . The position vectors of the points of a straight line through the origin form a one-dimensional sub - vector space of Euclidean space .

## Straight line through the origin in the plane

### definition

A straight line through the origin in the Euclidean plane is a straight line that passes through the origin of the coordinate system. In the coordinate form , a straight line through the origin consists of those points on the plane whose coordinates correspond to the straight line equation ${\ displaystyle (0,0)}$ ${\ displaystyle (x, y)}$ ${\ displaystyle ax + by = 0}$ where and are parameters that cannot both be zero. By solving this equation , one obtains the simpler form , provided that is ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle y}$ ${\ displaystyle b \ neq 0}$ ${\ displaystyle y = mx}$ with the slope . In this form, however, a straight line through the origin cannot run perpendicular to the x-axis. ${\ displaystyle m = - {\ tfrac {a} {b}}}$ ### Examples

Important examples of straight lines through the origin are the two coordinate axes with the straight line equations

${\ displaystyle y = 0}$ and   .${\ displaystyle x = 0}$ Other important examples of straight lines through the origin are the bisectors of the I. and III. as well as the II. and IV. quadrants with the straight line equations

${\ displaystyle xy = 0}$ and   .${\ displaystyle x + y = 0}$ ### Vector equations

Straight lines through the origin can also be described by vector equations . In parametric form , a straight line through the origin then consists of those points on the plane whose position vectors correspond to the equation ${\ displaystyle {\ vec {x}}}$ ${\ displaystyle {\ vec {x}} = s {\ vec {u}}}$ for meet. The position vectors of the points of a straight line through the origin are therefore scalar multiples of the direction vector . Alternatively, a straight line through the origin can also be in normal form using the normal equation ${\ displaystyle s \ in \ mathbb {R}}$ ${\ displaystyle {\ vec {u}}}$ ${\ displaystyle {\ vec {n}} \ cdot {\ vec {x}} = 0}$ can be specified. Here represents a normal vector of the straight line and the scalar product of the two vectors and . A straight line through the origin then consists of those points of the plane whose position vectors are perpendicular to the given normal vector. ${\ displaystyle {\ vec {n}}}$ ${\ displaystyle {\ vec {n}} \ cdot {\ vec {x}}}$ ${\ displaystyle {\ vec {n}}}$ ${\ displaystyle {\ vec {x}}}$ ### Plumb line

For each straight line through the origin there is a perpendicular straight line which also runs through the origin of the coordinates. This perpendicular line then has the coordinate representation

${\ displaystyle bx-ay = 0}$ or, if the slope of the starting line is, ${\ displaystyle m \ neq 0}$ ${\ displaystyle y = - {\ frac {1} {m}} x}$ .

A normal vector of the starting line is a direction vector of the perpendicular line and a direction vector of the starting line is a normal vector of the perpendicular line.

## Straight line through the origin in space

### definition

Vector equations can also be used to describe straight lines through the origin in higher-dimensional Euclidean spaces . In parametric form, a straight line through the origin with a direction vector then consists of those points in space whose position vectors give the equation ${\ displaystyle {\ vec {u}} \ in \ mathbb {R} ^ {n}}$ ${\ displaystyle {\ vec {x}} \ in \ mathbb {R} ^ {n}}$ ${\ displaystyle {\ vec {x}} = s {\ vec {u}}}$ for meet. As in the two-dimensional case, a straight line through the origin consists of all points in space whose position vectors are a scalar multiple of the directional vector of the straight line. However, a normal equation no longer describes a straight line in three-dimensional and higher-dimensional spaces, but a hyperplane . ${\ displaystyle s \ in \ mathbb {R}}$ ### Examples

In three-dimensional space, the three coordinate axes can be represented by the straight line equations

${\ displaystyle {\ vec {x}} = s {\ vec {e}} _ {1}, {\ vec {x}} = s {\ vec {e}} _ {2}}$ and   ${\ displaystyle {\ vec {x}} = s {\ vec {e}} _ {3}}$ to be specified. Where , and are the three standard unit vectors . ${\ displaystyle s \ in \ mathbb {R}}$ ${\ displaystyle {\ vec {e}} _ {1} = (1,0,0)}$ ${\ displaystyle {\ vec {e}} _ {2} = (0,1,0)}$ ${\ displaystyle {\ vec {e}} _ {3} = (0,0,1)}$ ### Distance of a point

The distance between a point with a position vector and a straight line through the origin with a direction vector is , where ${\ displaystyle {\ vec {v}}}$ ${\ displaystyle {\ vec {u}}}$ ${\ displaystyle | {\ vec {v}} - {\ vec {p}} |}$ ${\ displaystyle {\ vec {p}} = {\ frac {{\ vec {v}} \ cdot {\ vec {u}}} {{\ vec {u}} \ cdot {\ vec {u}}} } \, {\ vec {u}}}$ is the position vector of the plumb line , i.e. the orthogonal projection of the vector onto the straight line. ${\ displaystyle {\ vec {v}}}$ ### Vector space structure

The vectors in Euclidean space form a vector space , the so-called coordinate space . The set of position vectors of the points of a straight line through the origin forms a sub-vector space of Euclidean space

${\ displaystyle U = \ {{\ vec {x}} \ in \ mathbb {R} ^ {n} \ mid {\ vec {x}} = s {\ vec {u}} ~ {\ text {for} } ~ s \ in \ mathbb {R} \}}$ .

This sub-vector space is precisely the linear envelope of the direction vector of the straight line. The straight lines through the origin are the only one-dimensional sub - vector spaces of Euclidean space. ${\ displaystyle {\ vec {u}}}$ ### Straight line through the origin as a section

The two-dimensional subspaces of the three-dimensional Euclidean space are precisely the planes of origin . The intersection of two different planes of origin always results in a straight line through the origin, the direction vector of this straight line through the cross product

${\ displaystyle {\ vec {u}} = {\ vec {n}} _ {1} \ times {\ vec {n}} _ {2}}$ the normal vectors and the two planes of origin is given. In general, the -dimensional sub-vector spaces in the -dimensional Euclidean space are original hyperplanes and the intersection of such hyperplanes with linearly independent normal vectors always results in a straight line through the origin, the direction of which is given by the generalized cross product${\ displaystyle {\ vec {n}} _ {1}}$ ${\ displaystyle {\ vec {n}} _ {2}}$ ${\ displaystyle (n-1)}$ ${\ displaystyle n}$ ${\ displaystyle n-1}$ ${\ displaystyle {\ vec {n}} _ {1}, \ ldots, {\ vec {n}} _ {n-1}}$ ${\ displaystyle {\ vec {u}} = {\ vec {n}} _ {1} \ times \ cdots \ times {\ vec {n}} _ {n-1}}$ given is.