Coordinate shape

The coordinate form or coordinate equation is in mathematics a special form of a linear equation or plane equation . In the coordinate form, a straight line in the Euclidean plane or a plane in Euclidean space is described in the form of a linear equation . The unknowns of the equation are the coordinates of the points of the straight line or plane in a Cartesian coordinate system . The coordinate form is thus a special implicit representation of the straight line or plane.

Coordinate form of a straight line equation

Coordinate form of a straight line equation

presentation

In the coordinate form is a straight line in the plane by three real numbers , and described by a linear equation. A straight line then consists of those points whose coordinates the equation ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle (x,y)}$

${\displaystyle ax+by=c}$

fulfill. Here must be or not equal to zero. The numbers and are the components of the normal vector of the straight line. The distance of the straight line from the coordinate origin is indicated by. If the normal vector is normalized, i.e. a unit vector , then the distance is even . ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle {\vec {n}}=(a,b)}$ ${\displaystyle |c|/|{\vec {n}}|}$ ${\displaystyle |c|}$

example

In the picture above, the straight line equation is in coordinate form

${\displaystyle 1x+2y=6}$.

Every choice of which satisfies this equation, for example or , corresponds exactly to a straight line point. ${\displaystyle (x,y)}$ ${\displaystyle (0,3)}$ ${\displaystyle (4,1)}$

Special cases

• If is, the straight line runs parallel to the x-axis, and if is, parallel to the y-axis. ${\displaystyle a=0}$ ${\displaystyle b=0}$
• If is, the straight line is a straight line through the origin . ${\displaystyle c=0}$
• If is, the straight line equation is in intercept form ; the intersections are then and . ${\displaystyle c=1}$ ${\displaystyle (1/a,0)}$ ${\displaystyle (0,1/b)}$

calculation

From the normal form

The parameters of the coordinate form can be read directly from the normal form of a straight line equation with support vector and normal vector by multiplying the normal equation: ${\displaystyle {\vec {p}}}$ ${\displaystyle {\vec {n}}}$

${\displaystyle a=n_{1},~b=n_{2},~c=p_{1}n_{1}+p_{2}n_{2}}$.

If there is a straight line in Hessian normal form , the parameter can also be adopted from there. ${\displaystyle c}$

From the parametric form

From the parametric form of a straight line equation with support vector and direction vector , a normal vector of the straight line is first determined and then the parameters of the straight line in coordinate form as ${\displaystyle {\vec {p}}}$ ${\displaystyle {\vec {u}}}$ ${\displaystyle {\vec {n}}=(-u_{2},u_{1})}$

${\displaystyle a=-u_{2},~b=u_{1},~c=p_{1}a+p_{2}b}$.

From the two-point form

The parameters of the coordinate form are obtained from the two-point form of a straight line through the two points and by multiplying them out ${\displaystyle (x_{1},y_{1})}$ ${\displaystyle (x_{2},y_{2})}$

${\displaystyle a=y_{1}-y_{2},b=x_{2}-x_{1},c=x_{2}y_{1}-x_{1}y_{2}}$.

Coordinate form of a plane equation

Coordinate form of a plane equation

presentation

Similarly, a plane in three-dimensional space in the form of coordinates through four real numbers , , and described. A plane then consists of those points whose coordinates the equation ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle d}$ ${\displaystyle (x,y,z)}$

${\displaystyle ax+by+cz=d}$

fulfill. Here , or must be non-zero. The numbers , and are the components of the normal vector of the plane. The distance of the plane from the origin of coordinates is given by. If the normal vector is normalized, then the distance is even . ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle {\vec {n}}=(a,b,c)}$ ${\displaystyle |d|/|{\vec {n}}|}$ ${\displaystyle |d|}$

example

An example of a plane equation in coordinate form is

${\displaystyle 2x+3y-z=1}$.

Any choice of that satisfies this equation, for example or , corresponds to exactly one plane point. ${\displaystyle (x,y,z)}$ ${\displaystyle (1,0,1)}$ ${\displaystyle (0,1,2)}$

Special cases

• If is, the plane is parallel to the x-axis, if is, parallel to the y-axis, and if is, parallel to the z-axis. ${\displaystyle a=0}$ ${\displaystyle b=0}$ ${\displaystyle c=0}$
• If is, the plane is an origin plane . ${\displaystyle d=0}$
• If is, the plane equation is in intercept form ; the intersections are then , and . ${\displaystyle d=1}$ ${\displaystyle (1/a,0,0)}$ ${\displaystyle (0,1/b,0)}$ ${\displaystyle (0,0,1/c)}$

calculation

From the normal form

From the normal form of a plane equation with support vector and normal vector , the parameters of the plane can also be read in coordinate form by multiplying: ${\displaystyle {\vec {p}}}$ ${\displaystyle {\vec {n}}}$

${\displaystyle a=n_{1},~b=n_{2},~c=n_{3},~d=p_{1}n_{1}+p_{2}n_{2}+p_{3}n_{3}}$.

If a plane is in Hessian normal form , the parameter can also be adopted from there. ${\displaystyle d}$

From the parametric form

From the parameters the form of a plane equation with support vector and the two direction vectors , and a normal vector of the plane is first on the cross product determined and from the parameters of the plane in coordinate form as ${\displaystyle {\vec {p}}}$ ${\displaystyle {\vec {u}}}$ ${\displaystyle {\vec {v}}}$ ${\displaystyle {\vec {n}}={\vec {u}}\times {\vec {v}}}$

${\displaystyle a=u_{2}v_{3}-u_{3}v_{2},~b=u_{3}v_{1}-u_{1}v_{3},~c=u_{1}v_{2}-u_{2}v_{1},~d=p_{1}a+p_{2}b+p_{3}c}$.

In this way, a normal vector can also be determined from the three-point form of a plane equation, and then the coordinate form from this.

generalization

In general, a linear equation with unknowns describes a hyperplane in -dimensional Euclidean space. A hyperplane then consists of those points whose coordinates the equation ${\displaystyle n}$ ${\displaystyle x_{1},\dotsc ,x_{n}}$ ${\displaystyle n}$ ${\displaystyle (x_{1},\dotsc ,x_{n})}$

${\displaystyle a_{1}x_{1}+\dotsb +a_{n}x_{n}=b}$

fulfill. At least one of the parameters must be non-zero. ${\displaystyle a_{1},\dotsc ,a_{n}}$

literature

• Steffen Goebbels, Stefan Ritter: Understanding and applying mathematics . Springer, 2011, ISBN 978-3-8274-2762-5 .
• Peter Knabner, Wolf Barth: Linear Algebra: Fundamentals and Applications . Springer, 2012, ISBN 978-3-642-32186-3 .

Individual evidence

1. ↑ Peter Knabner, Wolf Barth: Lineare Algebra: Fundamentals and Applications . Springer, 2012, p. 41-42 .

Web links

• Convert plane from normal form to coordinate form. In: Serlo. Retrieved February 23, 2014 . 
• Convert plane from parametric form to coordinate form. In: Serlo. Retrieved February 23, 2014 .