Question

Prove that the level curves of the plane \(a x+b y+c z=d\) are parallel lines in the \(x y\) -plane, provided \(a^{2}+b^{2} \neq 0\) and \(c \neq 0\).

Short answer

Question: Prove that the level curves of the plane given by \(a x+b y+c z=d\) are parallel lines in the \(xy\)-plane under the conditions that \(a^{2}+b^{2} \neq 0\) and \(c \neq 0\). Answer: The level curves of the plane given by \(a x+b y+c z=d\) are parallel lines in the \(xy\)-plane, as their direction vectors are proportional (i.e., they have the same slope), which is a constant: \(\frac{dy}{dx} = \frac{-a}{b}\). This constant slope indicates parallelism of the level curves.

Step-by-step solution

Find the level curves for the given plane

A level curve is a cross-section of the 3D plane with a horizontal plane at a specific height \(z=k\). We can start by solving for \(z = k\) in the given equation: \(a x+b y+c z=d\) Since \(z=k\), we have \(a x+b y+c k=d\) Now, we will isolate \(y\) from this equation: \(y = \frac{d-ax-ck}{b}\) We will use this equation to analyze the direction vectors of our level curves.

Analyze the direction vector of the level curves

Since the equation we derived is in the form of \(y = \frac{d-ax-ck}{b}\), we can see that it represents lines in the \(xy\)-plane. To prove these lines are parallel, we need to show that their direction vectors are proportional (i.e., they have the same slope). The slope of the line is given by the rate of change of \(y\) with respect to \(x\), which is the derivative of \(y\) with respect to \(x\). We can find this by taking the derivative of \(y\): \(\frac{dy}{dx} = \frac{-a}{b}\) Since \(a^{2} + b^{2} \neq 0\), we know that \(b \neq 0\), which means the derivative exists and is constant.

Prove that the level curves' direction vectors are parallel

Since \(\frac{dy}{dx} = \frac{-a}{b}\) is a constant which neither depends on \(x\) nor is affected by the horizontal plane's height \(k\), this shows that the slope of the level curves is constant for all values of \(k\). Thus, the direction vectors for all level curves are proportional, proving that these level curves are parallel lines in the \(xy\)-plane.

Key concepts

Plane Equation

In the context of mathematics, a plane equation can be thought of as a flat, two-dimensional surface extending infinitely in three-dimensional space. The general equation for a plane in three dimensions is given by: \( ax + by + cz = d \). This formula uses three variables: \(x\), \(y\), and \(z\), which represent coordinates in the 3D system. Here:

• \(a\), \(b\), and \(c\) are coefficients that together determine the orientation of the plane,
• and \(d\) represents a constant which shifts the plane in space.

A plane equation can describe various planes depending on the values of its coefficients. If two of these coefficients are zero, the plane reduces to align with one of the coordinate planes, such as the xy-plane, yz-plane, or zx-plane. Planes intersecting one of these planes at a certain height slice through along what are known as level curves. In proving the properties of these level curves, it's necessary to modify the plane equation based on conditions like fixing a coordinate, such as setting \(z\) to a constant value \(k\).

Direction Vector

The direction vector of a line characterizes the direction in which the line extends and is particularly useful in analyzing lines in the context of level curves in planes. For level curves, which are essentially lines in the xy-plane derived from fixing \(z\) in a plane equation, finding the direction vector helps clarify their orientation and alignment.When examining the line equation derived from the plane equation, such as \( ax + by = d - ck \), converting it into the slope-intercept form (\(y = mx + b\)) allows the extraction of the direction vector. Here, \(m\) is the slope, and the corresponding direction vector is \( \langle 1, m \rangle \).In our example, we derived:

• The direction vector for the level curves is essentially \( \langle 1, -\frac{a}{b} \rangle \).

A key observation is that if two lines share a direction vector, they are parallel. This consistent direction vector confirms that all level curves extracted from the plane at varying heights are indeed parallel lines, as their direction does not vary with changes in \(k\).

Slope of a Line

The slope of a line in mathematics is a measure of its steepness or incline. It's calculated as the change in the vertical direction (\(y\)) over the change in the horizontal direction (\(x\)), typically denoted as \(m\) in line equations like \(y = mx + b\).For lines characterized by plane level curves, understanding the slope is critical in proving parallelism. For instance, when given a line equation derived as \(y = \frac{d-ax-ck}{b}\), the slope is found by identifying the coefficient of \(x\). In this setup:

• The slope \( \frac{-a}{b} \) emerges from position \(x\) division in the equation \(y = \frac{d-ax-ck}{b}\).

The fact the slope is constant across all level curves shows that each line's incline is the same, resulting in parallel lines. When analyzing geometric figures, constant slopes highlight the invariant linear relationships between coordinates across dimensional projections, particularly in 3D to 2D reductions like level curves.

3D Coordinate Systems

A 3D coordinate system is an extension of the 2D system, providing an additional axis for spatial navigation. Whereas a 2D system uses just \(x\) and \(y\) axes to define a point's position on the plane, the 3D system introduces a \(z\)-axis, allowing for representation of positions in volumetric space.Key aspects include:

• The \(x\)-axis represents the breadth.
• The \(y\)-axis represents height.
• The \(z\)-axis adds depth.

In 3D space, points are represented by coordinates \((x, y, z)\), indicating their location relative to the origin.Plane equations, like \( ax + by + cz = d \), further describe infinite flat surfaces within this 3D space. To analyze sections of these surfaces, changing a variable (e.g., setting \(z = k\)) reduces the scenario to a 2D system, often expressed through level curves or sections that become "slices" of the 3D figure. This simplifies analyzing spatial properties and solving geometry-based problems by viewing them through the simpler 2D lens.