Problem 12 - Thomas Calculus

Question

Find parametric equations for the lines in Exercises 1–12. The \(z\) -axis

Step-by-Step Solution

Answer

The parametric equations for the z-axis are \( x = 0 \), \( y = 0 \), and \( z = t \).

Step 1: Understand the Equation of the z-axis

The z-axis in a 3D coordinate system is a line parallel to the z-coordinate axis. Points on this line have their x and y coordinates both equal to zero. Therefore, any point on the z-axis can be written as (0, 0, z), where z is any real number.

Step 2: Define the Direction Vector

The direction vector of the z-axis indicates how the line extends through space. Since movement occurs only along the z direction, the direction vector is (0, 0, 1). This means there is no movement in the x or y directions, only in the z direction.

Step 3: Write the Parametric Equations

Parametric equations describe a line by giving separate equations for each coordinate as a function of a parameter, commonly denoted as t. For the z-axis:- The x-coordinate remains constantly zero: \( x = 0 \).- The y-coordinate also remains constantly zero: \( y = 0 \).- The z-coordinate varies linearly with the parameter t: \( z = t \).

Step 4: Finalize the Parametric Form

Based on the analysis, the parametric equations for the z-axis are:\[x(t) = 0 \]\[y(t) = 0 \]\[z(t) = t \]These equations indicate that points on the line move up or down the z-axis without deviating along the x or y directions.

Key Concepts

3D Coordinate System

A 3D coordinate system extends the conventional 2D plane into three dimensions. These dimensions are represented by the x, y, and z axes. The 3D coordinate system forms the basis for understanding the position of any object or point in three-dimensional space. Each point in this space is defined by an ordered triplet \(x, y, z\), where these values correspond to coordinates on their respective axes.
A key feature of this system includes:

• **Origin:** The point (0, 0, 0) where all three axes intersect.
• **Axes:** The x-axis runs horizontally, the y-axis vertically, and the z-axis runs perpendicular to both, usually depicted as coming out of or going into the page.
• **Quadrants:** Instead of four quadrants as in a 2D system, the 3D system is divided into eight octants, offering a more complex spatial division.

Understanding the 3D coordinate system is foundational for visualizing geometric concepts like lines, planes, and curves in three-dimensional space.

Direction Vector

In the 3D context, a direction vector describes the orientation and direction of a line. For example, a line through three-dimensional space is defined by both a point on the line and a direction vector. This direction vector embodies the direction in which the line extends.

• The components of the direction vector—let's say (a, b, c)—indicate how much we "move" in the x, y, and z directions, respectively, as we traverse the line.
• For the z-axis with a direction vector of (0, 0, 1), our movement is exclusively along the z direction with zero movement in the x or y directions.
• Direction vectors are essential when defining parametric equations because they describe how the parameter t influences each coordinate.

Choosing the right direction vector is crucial for accurately representing the line in equations, affecting how it is drawn or visualized in the 3D coordinate system.

z-axis

The z-axis is an integral part of the 3D coordinate system, representing the third dimension. It is perpendicular to both the x and y axes. Moving along the z-axis, the x and y coordinates of any point remain constant—specifically at zero when describing the z-axis line itself.
Attributes of the z-axis include:

• **Simplicity in Parametrics:** Since the z-axis is aligned with the z-axis itself, its parametric equations are straightforward, with x(t) = 0, y(t) = 0, and z(t) = t.
• **Vertical Alignment:** It represents vertical movement in 3D space, often used to describe anything from depth to height depending on its application context.
• **Symmetry with Other Axes:** Being an axis of the Cartesian coordinate system, it mirrors the characteristics of the x and y axes, providing uniformity across dimensions.

Grasping the role of the z-axis is vital for understanding spatial orientation and navigating three-dimensional mathematical frameworks.