Point test

In the point sample , a computational decision is made as to whether a point lies in a given point set , i.e. whether there is an incidence . Different point sets are possible:

There is a point

• on a function graph in an xy coordinate system ?
• on a straight line in the three-dimensional coordinate system?
• on a plane in the three-dimensional coordinate system?

Procedure

A point test is performed by plugging the coordinates of the point into the equation of the point set. If the point satisfies the equation, i. H. if a true statement arises, the point lies in the point set. If a wrong statement arises, the point is not in the point set.

Thus it is possible to check at the end of an invoice whether z. B. a calculated intersection of two straight lines actually lies on both straight lines.

Examples

Linear function

Does the point lie on the straight line with the function  equation? ${\displaystyle P(4|7)}$ ${\displaystyle g:\ y=2x-5}$

The x-coordinate of the point P is used for and the y-coordinate of the point is used. . This is not a true statement, so point P is not on the graph. ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle 7=2\cdot 4-5}$

Line equation in parametric form

Does the point lie on the straight line h with the parametric equation  ? ${\displaystyle Q(8|3|5)}$ ${\displaystyle h:{\vec {x}}={\begin{pmatrix}2\\0\\4\end{pmatrix}}+\lambda \cdot {\begin{pmatrix}3\\-1\\2\end{pmatrix}},\lambda \in \mathbb {R} }$

The position vector of point Q is used for the vector and resolved line by line for the parameter . Since this results in the first line but also delivers the second line , there is a contradiction . Thus the point Q does not lie on the straight line h, short . ${\displaystyle {\vec {x}}}$ ${\displaystyle \lambda }$ ${\displaystyle \lambda =2}$ ${\displaystyle \lambda =-3}$ ${\displaystyle Q\notin h}$

Plane equation in coordinate form

Is the point on the plane with the coordinate equation  ? ${\displaystyle R(2|1|11)}$ ${\displaystyle {E:\ 3x_{1}+7x_{2}-x_{3}=2}}$

For , and insert the coordinates of the point R. . This is a true statement, so the point R is on the plane, in short . ${\displaystyle x_{1}}$ ${\displaystyle x_{2}}$ ${\displaystyle x_{3}}$ ${\displaystyle {\ 3\cdot 2+7\cdot 1-11=2}}$ ${\displaystyle R\in E}$

Other uses

The point sample can also be used to determine a straight line equation g if a point P of the straight line and its slope are known. ${\displaystyle m}$

Approach: ${\displaystyle g:\ y=m\cdot x+c}$

The y-axis intercept is now determined by performing the "point test" for point P and resolving for. This is an alternative to the point slope form . ${\displaystyle c}$ ${\displaystyle c}$

The point probe, can thus three points of are given, for the determination of a quadratic equation, and a function f terms are used, having a parabola as a graph. The general quadratic function is: ${\displaystyle P_{1},P_{2},P_{3}}$ ${\displaystyle R^{2}}$

${\displaystyle f:\ f(x)=ax^{2}+bx+c}$

Now the point test is carried out for each of the points P1, P2, P3 and a linear system of equations with the unknowns a, b, c is obtained. After solving the problem, the functional term of the function f is obtained, which turns into a true statement for the coordinates of P1, P2, P3.