Problem 12 - University Calculus: Early Transcendentals

Question

Find parametric equations for the lines. The z-axis

Step-by-Step Solution

Answer

Parametric equations: \(x = 0\), \(y = 0\), \(z = t\).

Step 1: Understand the Problem

Since we are asked to find parametric equations for the z-axis, we need equations that represent points on this vertical line. The z-axis can be described as passing through the origin and having no changes in the x or y coordinates, just the z coordinate changing.

Step 2: Define Parameters for the Line

For parametric equations, we need to express each coordinate in terms of a single parameter, say \(t\). Typically, we can let \(x\), \(y\), and \(z\) be expressed as functions of \(t\).

Step 3: Express x and y in Terms of the Parameter

Since the line is the z-axis, both \(x\) and \(y\) are always 0 regardless of the value of \(z\). Thus, the parametric equations for \(x\) and \(y\) will be: \(x = 0\) and \(y = 0\).

Step 4: Express z in Terms of the Parameter

The z-axis allows the z-value to change freely, so we can represent this by letting \(z = t\). The parameter \(t\) can represent the entire range of real numbers, covering all points on the z-axis.

Step 5: Compile the Parametric Equations

Now we have a complete set of parametric equations for the z-axis:\[ x(t) = 0 \]\[ y(t) = 0 \]\[ z(t) = t \]

Key Concepts

z-axis

The z-axis is a crucial part of the three-dimensional coordinate system. It is one of the three axes, along with the x-axis and y-axis, that allow us to define the location of points in a 3D space. The z-axis is typically represented as a vertical line passing through the origin, where the values of x and y are both zero.
Its direction is perpendicular to both the x and y planes, making it unique in defining the elevation or depth of a point.

• Points on the z-axis have the form (0, 0, z), where z can take any real number value.
• It is often used in mathematical problems involving 3D space to describe motion or position that is purely vertical.

This makes the z-axis an essential axis for understanding and constructing three-dimensional geometric shapes and graphs.

coordinate system

A coordinate system is a framework used to define the position of points in space. There are various types of coordinate systems, but the most common in everyday applications are Cartesian coordinates. In the Cartesian coordinate system, a point in a plane is identified by an ordered pair of numbers (x, y), and in 3D space, by a triplet (x, y, z).
This system is simple and intuitive as it describes geometric locations with numerical values along perpendicular axes, typically making up a grid where each number represents a distance from the origin.

• It's widely used due to its simplicity in mapping a physical space onto a numerical grid.
• In a 3D Cartesian system, each axis (x, y, z) is at right angles to each other, facilitating easy computation of distances and angles.

Being familiar with the coordinate system is vital for students as it underpins much of vector calculus and geometric calculations used in science and engineering.

parametrization

Parametrization involves expressing a set of points through a parameter, enabling the description of curves and surfaces in terms of a single variable. This approach is invaluable in simplifying the representation of geometric shapes and motion along paths.
For example, if you have a line in space, you can describe its path by a set of parametric equations using a parameter, often denoted by \( t \). The essence of parametrization is to

describe

each component of the figures (x, y, z coordinates) as continuous functions of \( t \), which parameterizes the curves along certain directions or surfaces.

• By adjusting the parameter value, you can trace the path or the shape completely.
• Parametrization is essential in physics for describing motion and calculating trajectories.

These representations make it easier to solve complex geometrical and physical problems, translating intricate shapes into manageable calculations.

mathematical representation

Mathematical representation refers to the different ways in which mathematical ideas and structures can be expressed. This could be through equations, graphs, models, diagrams, or verbal descriptions. In mathematics, visual and algebraic representation play significant roles in problem-solving and conveying complex concepts.

• Equations are a form of representation that use symbols and variables to denote relationships and operations.
• Graphs can give visual insight into the behavior of functions and their respective variables.

In the context of parametric equations, mathematical representations are powerful in translating real-world scenarios into analyzable data through visual and symbolic formats. Understanding these representations allows for effective navigation between different mathematical ideas and practical applications, enhancing both computation and interpretation.