91Ó°ÊÓ

In mathematics, **three-dimensional (3D) geometry** deals with objects that have three components: length, width, and height. This geometry allows us to represent objects in the space around us, such as a cube or a sphere.
3D space is represented using three coordinates (x, y, z), each describing a dimension. These coordinates help specify any point in space uniquely.
In this exercise, we are dealing with a 3D linear function: \(z = c + mx + ny\). This represents a plane in 3D space, which can be visualized as a flat surface extending infinitely in the directions determined by its equation.

• The term \(z\) represents the vertical dimension, often seen as height.
• \(x\) and \(y\) represent the horizontal dimensions, akin to length and width.

This type of line creates surfaces, such as planes, that can be used for understanding the relationship between variables in a three-dimensional context.

A **linear equation** is any equation that makes a straight line when it is graphed. In one dimension, it's represented by \(y = mx + c\) where \(m\) is the slope, and \(c\) is the y-intercept.
When extended into three dimensions, it turns into a linear plane: \(z = c + mx + ny\).
Here, it expresses the relationship between three variables.

• \(m\) links changes in \(x\) to changes in \(z\) - known as the gradient in the direction of the x-axis.
• \(n\) links changes in \(y\) to changes in \(z\) - known as the gradient in the direction of the y-axis.

Linear equations are fundamental in mathematics as they are used to model a large variety of real-world situations, including this representation of a plane.

**Points in space** are described using ordered triples, \((x, y, z)\), which represent a unique location in three-dimensional geometry.
The concept of points is foundational in geometry, as every line or plane can be defined or determined by points.
In the provided exercise:

• The point \((0,0,0)\) is the origin, offering a reference point where the x, y, and z-axes intersect.
• \((0,2,-1)\) lies on the y-axis and beneath the origin, showing an effect on the \(z\) coordinate.
• \((-3,0,-4)\) extends along the x-axis and below the origin, influencing both \(x\) and \(z\).

Proximity to each other, these points help construct the plane described by the linear equation. By substituting these points in the equation \(z = c + mx + ny\), we derived the specific expressions for \(m\) and \(n\) that reflect how the plane interacts with the x and y dimensions.