Straight line through the origin

Straight line through the origin in the Euclidean plane

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

${\displaystyle x-y=0}$   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

A straight line through the origin in three-dimensional 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

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

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

A straight line through the origin as the intersection of two planes of origin

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.

See also

• Proportionality
• Projective space

literature

• Kenneth Eriksson, Donald Estep, Claes Johnson: Applied Mathematics: Body and Soul 1 . Springer, 2006, ISBN 3-540-35006-3 .
• Mike Scherfner, Torsten Volland: Mathematics for the first semester . Springer, 2012, ISBN 3-8274-2505-0 .