Straight line equation

Straight through the two points and in a Cartesian coordinate system ${\displaystyle P}$ ${\displaystyle Q}$

A straight line equation is an equation in mathematics that uniquely describes a straight line . The straight line consists of all the points whose coordinates satisfy the equation.

The figure shows a straight line through two given points and in a Cartesian coordinate system . Due to two different points, there is always exactly one straight line in Euclidean geometry . ${\displaystyle P}$ ${\displaystyle Q}$

Straight lines in the plane

Coordinate equations

In a Cartesian coordinate system, two numbers and coordinates are assigned to each point on the plane . One writes or . An equation with the variables and then describes a set of points in the plane, namely the set of all points whose - and -coordinates satisfy the equation. The spelling ${\displaystyle P}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle P(x|y)}$ ${\displaystyle P=(x,y)}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle y}$

${\displaystyle g\colon \ y=2x}$

means, for example, that the straight line consists of all points that satisfy the equation . The corresponding set notation is ${\displaystyle g}$ ${\displaystyle (x,y)}$ ${\displaystyle y=2x}$

${\displaystyle g=\{(x,y)\mid y=2x\}}$.

Straight lines are characterized by the fact that the associated straight line equation is a linear equation . There are a number of different forms of representation for such equations.

Main or normal form

Straight line with slope m and y-axis intercept n

→ Main article : Linear function

Any straight line that is not parallel to the y-axis is the graph of a linear function

${\displaystyle f(x)=m\cdot x+n}$,

where and are real numbers . The associated straight line equation then reads ${\displaystyle m}$ ${\displaystyle n}$

${\displaystyle y=m\cdot x+n}$.

The parameters and the straight line equation have a geometric meaning. The number is the slope of the straight line and corresponds to the vertical cathetus of the slope triangle whose horizontal cathetus length has. The number is the y-axis intercept , i.e. the straight line intersects the y-axis at the point . If , then the straight line runs as the straight line through the origin of the coordinates and the associated function is then a proportionality . The straight line with the equation is obtained from the straight line with the equation by shifting it in the direction of the y-axis. This shift is up when it is positive and down when it is negative. ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle m}$ ${\displaystyle 1}$ ${\displaystyle n}$ ${\displaystyle (0,n)}$ ${\displaystyle n=0}$ ${\displaystyle y=m\cdot x+n}$ ${\displaystyle y=m\cdot x}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$

Straight lines that run parallel to the y-axis are not function graphs. They can be expressed by an equation of form

${\displaystyle x=a}$

represent, where is a real number. Such a straight line intersects the x-axis at the point . ${\displaystyle a}$ ${\displaystyle (a,0)}$

Two-point form

Slope triangles of a straight line

→ Main article : Two-point form

Passing the line through the two points and , wherein and are different, then the slope can be of the straight line by means of the difference quotient by ${\displaystyle (x_{1},y_{1})}$ ${\displaystyle (x_{2},y_{2})}$ ${\displaystyle x_{1}}$ ${\displaystyle x_{2}}$ ${\displaystyle m}$

${\displaystyle m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}}$

be calculated. According to the theorem of rays , any other point on the straight line can be selected instead of the point without changing the slope. This results in the two-point form ${\displaystyle (x_{2},y_{2})}$ ${\displaystyle (x,y)}$

${\displaystyle {\frac {y-y_{1}}{x-x_{1}}}={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}}$

or equivalent, by solving the equation for , ${\displaystyle y}$

${\displaystyle y={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}\cdot (x-x_{1})+y_{1}}$

and thus

${\displaystyle y=\underbrace {\left({\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}\right)} _{m}x+\underbrace {\frac {y_{1}x_{2}-y_{2}x_{1}}{x_{2}-x_{1}}} _{n}}$.

Point slope shape

Point slope form of a straight line equation

→ Main article : Point slope form

A straight line through the point with the slope is described by the following equation: ${\displaystyle (x_{1},y_{1})}$ ${\displaystyle m}$

${\displaystyle y-y_{1}=m\cdot (x-x_{1})}$.

This formula can also be used if two points are known, but you do not want to explicitly determine the intersection with the y-axis (mentioned above ). ${\displaystyle n}$

Coordinate shape

→ Main article : coordinate shape

The coordinate form of the straight line equation in the plane is

${\displaystyle ax+by=c}$,

where and cannot both be 0. ${\displaystyle a}$ ${\displaystyle b}$

By solving the equation for (if ) you get the explicit form. The coordinate shape has the advantage that it is symmetrical in and . So no direction of the straight line is preferred. Straight lines that are parallel to the y-axis do not play a special role. ${\displaystyle y}$ ${\displaystyle b\neq 0}$ ${\displaystyle x}$ ${\displaystyle y}$

Intercept shape

Intercept form of a straight line equation

→ Main article : Intercept shape

A special form of coordinate form is the intercept form. If the straight line intersects the x-axis at the point and the y-axis at the point , where and are not zero, the straight line equation can be expressed in the form ${\displaystyle (x_{0},0)}$ ${\displaystyle (0,y_{0})}$ ${\displaystyle x_{0}}$ ${\displaystyle y_{0}}$

${\displaystyle {\frac {x}{x_{0}}}+{\frac {y}{y_{0}}}=1}$

write. This form is called the intercept form of the straight line equation with the x-intercept and the y-intercept . If the equation is solved for , the explicit form results ${\displaystyle x_{0}}$ ${\displaystyle y_{0}}$ ${\displaystyle y}$

${\displaystyle y=-{\frac {y_{0}}{x_{0}}}\cdot x+y_{0}}$,

where the ratio corresponds to the slope of the straight line. ${\displaystyle -{\tfrac {y_{0}}{x_{0}}}}$ ${\displaystyle m}$

Vector equations

There is also the possibility of describing a straight line with the help of vector calculation . Instead of the points, one considers their position vectors . The position vector of a point is usually referred to as. ${\displaystyle {\overrightarrow {OP}}}$ ${\displaystyle P=(p_{1},p_{2})}$ ${\displaystyle {\vec {p}}={\tbinom {p_{1}}{p_{2}}}}$

Parametric shape

Parametric form of a straight line equation

→ Main article : parametric shape

In the parametric form, no condition is formulated that the coordinates of the points must meet in order for them to lie on the straight line, but the points of the straight lines are displayed as a function of a parameter. A point on the straight line corresponds to each value of the parameter. If the parameter runs through all real numbers, all points of the straight line are obtained. In the parametric form, a straight line is shown

${\displaystyle {\vec {x}}={\vec {p}}+s\,{\vec {u}}}$

or advertised

${\displaystyle {x_{1} \choose x_{2}}={p_{1} \choose p_{2}}+s\,{u_{1} \choose u_{2}}}$.

Here is the position vector of a fixed point of the straight line, the direction vector of the straight line and a number that indicates how long it is counted in this direction. The parameter forms the coordinate of an affine coordinate system on the straight line, that is, the straight line is numbered with the values ​​of , with the zero point at . ${\displaystyle {\vec {p}}}$ ${\displaystyle {\vec {u}}}$ ${\displaystyle s}$ ${\displaystyle s}$ ${\displaystyle s}$ ${\displaystyle (p_{1},p_{2})}$

Normal form

Normal form of a straight line equation

→ Main article : Normal form

With a normal vector that is at right angles to the straight line, the straight line can be written in normal form: ${\displaystyle {\vec {n}}}$

${\displaystyle {\vec {n}}\cdot ({\vec {x}}-{\vec {p}})=0}$.

This again contains the position vector of a straight line point and the scalar product of two vectors. If a direction vector is a straight line, then is a normal vector of the straight line. In the Hessian normal form ${\displaystyle {\vec {p}}}$ ${\displaystyle \cdot }$ ${\displaystyle {\tbinom {u_{1}}{u_{2}}}}$ ${\displaystyle {\tbinom {-u_{2}}{u_{1}}}}$

${\displaystyle {\vec {n}}_{0}\cdot {\vec {x}}=d}$

a straight line is described by a standardized and oriented normal vector and the distance from the origin of coordinates . ${\displaystyle {\vec {n}}_{0}}$ ${\displaystyle d}$

Straight lines in space

Representation of a straight line

Straight lines in space cannot be represented in the normal form because they have neither axis intercepts nor a clearly defined normal vector (there are infinitely many directions perpendicular to a straight line in space). The parametric form presented above is common

${\displaystyle {\vec {x}}={\vec {p}}+s\,{\vec {u}}}$,

where , and are now vectors in space. With the help of the vector product , another, parameter-free straight line form can be constructed, the determinant form ${\displaystyle {\vec {x}}}$ ${\displaystyle {\vec {p}}}$ ${\displaystyle {\vec {u}}}$

${\displaystyle {\vec {u}}\times {\vec {x}}-{\vec {u}}\times {\vec {p}}={\vec {0}}}$.

Here again the position vector of a fixed point is the straight line and the direction vector of the straight line. Since the difference between the position vector of any point on the straight line and the support vector must be collinear to the direction vector ( i.e. pointing in the same or in the opposite direction), the vector product of the two always results in the zero vector : ${\displaystyle {\vec {p}}}$ ${\displaystyle {\vec {u}}}$ ${\displaystyle {\vec {x}}-{\vec {p}}}$ ${\displaystyle {\vec {x}}}$ ${\displaystyle {\vec {p}}}$ ${\displaystyle {\vec {u}}}$

${\displaystyle {\vec {u}}\times ({\vec {x}}-{\vec {p}})={\vec {0}}}$.

The equation applies to every vector that is the position vector of a point on the straight line; in all other cases the zero vector does not result. If is a unit vector , then corresponds to ${\displaystyle {\vec {x}}}$ ${\displaystyle {\vec {u}}}$

${\displaystyle |{\vec {u}}\times {\vec {p}}|}$

exactly the distance of the straight line from the origin.

See also

• Plane equation

literature

• Manfred Leppig: Learning Levels in Mathematics . Girardet 1981, ISBN 3-7736-2005-5 , pp. 61-74
• Ilja Nikolajewitsch Bronstein , Konstantin Adolfowitsch Semendjajew : Taschenbuch der Mathematik . Harri Deutsch Verlag, 24th edition 1989, ISBN 3-87144-492-8 , p. 219
• Helmuth Preckur: Linear Algebra and Analytical Geometry . Mentor Verlag (Mentor-Lernhilfe Volume 50), Munich 1983, ISBNM3-580-64500-5, pp. 72-85, 106-114

Remarks

1. ↑ The parameter is also referred to as , or in the literature . In Austria you usually write . ${\displaystyle n}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle t}$ ${\displaystyle f(x)=k\cdot x+d}$

Web links

Commons : Linear Equations  - collection of images, videos and audio files

Commons : Linear functions  - collection of images, videos and audio files

• Straight line equation . In: Serlo .