Question

a. Show that there is a line through any pair of points in the plane. b. Generalize and show that there is a plane $ax+by+$ $cz+d=0$ through any three points in space.

Short answer

Any two points determine a line, and any three non-collinear points determine a plane.

Step-by-step solution

Understanding Points and Lines

In a plane, any two distinct points can be connected with a line. A line is determined by its two distinct points, and it can extend infinitely in both directions. We need to verify this by considering the basic equation of a line in a two-dimensional plane.

Determining Equation of a Line

Consider two points in the plane, say $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$. The general form of the equation of a line passing through these points is $y-y_1=m(x-x_1)$, where $m$ is the slope, $m=\frac{y_2-y_1}{x_2-x_1}$. Since $x_{1}eq{x}_2$, the slope $m$ is well-defined.

Conclusion for Part A

By setting up an equation based on two distinct points, we have proven that there is always a line determined by these points as long as $x_{1}eq{x}_2$ or $y_{1}eq{y}_2$. Therefore, any pair of points can form a line in a two-dimensional plane.

Understanding Points and Planes

In three-dimensional space, a plane is determined by any three non-collinear points. We need to determine the equation of such a plane by considering its general form $ax+by+cz+d=0$.

Setting the Plane Equation

Consider three non-collinear points $P_1(x_1,y_1,z_1)$, $P_2(x_2,y_2,z_2)$, and $P_3(x_3,y_3,z_3)$. To find $a,b,c,\text{and }d$, ensure the equation satisfies all three points by substituting them into the plane equation. Solve the resulting system of equations to determine these coefficients uniquely.

Verification of Plane Equation

Verify the constructed plane equation by checking that substituting $x_1,y_1,z_1$; $x_2,y_2,z_2$; and $x_3,y_3,z_3$ into it results in zero. This confirms that the plane indeed passes through the given three points.

Conclusion for Part B

Since three non-collinear points uniquely determine an equation of the form $ax+by+cz+d=0$, we've shown that there exists a plane passing through any such trio of points.

Key concepts

Equation of a Line

In geometry, a line is a one-dimensional figure that is perfectly straight and extends infinitely in both directions. It is often depicted by a line with two arrowheads in drawings. In computer graphics, the equation of a line is a fundamental concept that helps in determining the path between two distinct points on a plane.

To find the equation of a line in a two-dimensional space, you typically begin with two points, let’s call them $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$.

• The slope $m$ is calculated using the formula: $$m=\frac{y_2-y_1}{x_2-x_1}$$ This gives the rate at which the $y$ coordinate changes with respect to $x$.


• Once the slope is determined, the line’s equation can be written in the form: $$y-y_1=m(x-x_1)$$ This is known as the point-slope form of the equation of a line.

By ensuring that each point satisfies this line equation, you confirm that the line indeed passes through these points. This principle establishes the foundational relationship in geometry where any two points can form a line.

Equation of a Plane

Unlike a line, a plane extends infinitely in two dimensions. It is like the surface of a piece of paper with no thickness, stretching infinitely wide and long. In three-dimensional geometry, the equation of a plane is a crucial tool used in computer graphics to define flat surfaces.

To define a plane, at least three non-collinear points are needed. Non-collinear means they do not all lie on the same straight line. Consider three such points $P_1(x_1,y_1,z_1)$, $P_2(x_2,y_2,z_2)$, and $P_3(x_3,y_3,z_3)$. The equation of a plane can be written as:

• General form: $$ax+by+cz+d=0$$ Where $a$, $b$, $c$, and $d$ are constants that define the plane.


• By substituting these points into the plane equation, you get a system of equations. Solving it will give you the specific values of $a$, $b$, $c$, and $d$.

Once these values are known, the equation $ax+by+cz+d=0$ describes a unique plane in 3D space, ensuring all three given points lie on it. This verifies that a plane can indeed be determined by three points.

Three-Dimensional Space

Three-dimensional space (3D space) is a geometric model of the physical universe in which we live. It adds depth to the traditional two-dimensional world of length and width, introducing a third dimension known as height or depth.

3D space is described by three coordinates, typically referred to as $x$, $y$, and $z$. Each of these coordinates contributes to the location of a point within the space. The concepts of lines and planes extend naturally from 2D space into 3D space:

• Lines in 3D: Just like in 2D, a line in 3D is defined by a point plus a direction. For instance, if you know a point on the line and the direction it extends, you can write the equation of this line using vectors or parametric form.


• Planes in 3D: Planes are crucial in 3D space as they define flat surfaces on which other objects may lie. The normal vector, perpendicular to the plane, is often used to describe its orientation.

Understanding 3D space is fundamental in fields like computer graphics, where it is essential for modeling and rendering realistic scenes. Three-dimensional concepts allow for the simulation of real-world environments and objects on a computer screen.