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- Points, Lines, and Planes
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We can recall that a plane is a two-dimensional surface made up of points that extends infinitely in all directions and there exists exactly one plane through any three noncollinear points. This word, noncollinear, simply means that the three points don’t lie on a straight line.
I understand that if I take one point or any number of collinear points, then I can draw infinite planes just by rotating around the line that connects these points, but why do we need 3 non collinear points to define a plane, why not more?
A projective plane is a set of points and subsets called lines that satisfy the following four axioms: P1. Any two distinct points lie on a unique line. P2. Any two lines meet in at least one point. P3. Every line contains at least three points. P4. There exist three noncollinear points.
Plane Equation for Three Non-Collinear Points. Given three non-collinear points in R 3 space, there is a unique plane that passes through these points: P 1, P 2, and P 3. $$ P_1 \begin{pmatrix} x_1 \\ y_1 \\ z_1 \end{pmatrix} \:\:\: P_2 \begin{pmatrix} x_2 \\ y_2 \\ z_2 \end{pmatrix} \:\:\: P_3 \begin{pmatrix} x_3 \\ y_3 \\ z_3 \end{pmatrix} $$
16 sty 2023 · In both cases, to find the equation of the plane that contains those two lines, simply pick from the two lines a total of three noncollinear points (i.e. one point from one line and two points from the other), then use the technique above, as in Example 1.24, to write the equation.
If the points (푥, 1), (3, 푥), and (0, −2) are collinear, use determinants to find all the possible values of 푥. Round your answers to two decimal places.
