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 are \(x(t) = 0\), \(y(t) = 0\), \(z(t) = t\).

Step 1: Understand the Problem

We need to find parametric equations for a line. Specifically, the exercise asks for the parametric equations for the line that is the z-axis.

Step 2: Identify the Direction Vector

The z-axis is a vertical line running parallel to the z-coordinate of the coordinate system. Therefore, the direction vector of the z-axis can be represented as \(\langle 0, 0, 1 \)\.

Step 3: Choose a Point on the Line

A point on the z-axis can be any point where the x and y coordinates are both zero. For simplicity, we can choose the point (0, 0, 0) which is the origin.

Step 4: Write the Parametric Equations

Using the point (0, 0, 0) and the direction vector \(\langle 0, 0, 1 \)\, the parametric equations of the z-axis are:- \(x(t) = 0\)- \(y(t) = 0\)- \(z(t) = t\), where \(t\) is a parameter.

Key Concepts

Direction Vectors

Direction vectors are essential in understanding the orientation of a line in space. They tell us in which direction the line extends by representing the line's slope or inclination. Essentially, a direction vector indicates the movement needed along each axis to travel across the line.In 3D space, a vector composed as \( \langle a, b, c \rangle \) indicates a move by \(a\) units in the x-direction, \(b\) units in the y-direction, and \(c\) units in the z-direction. For example, the direction vector of the z-axis is \( \langle 0, 0, 1 \rangle \), meaning it moves 0 units in both x and y directions but 1 unit in the z-direction. This reflects the fact that the z-axis climbs vertically without deviating to the left or right.Understanding direction vectors is instrumental as they form the backbone of parametric equations for lines. It helps in visualizing how a line behaves in 3D space by showing how the line stretches along each coordinate axis.

Coordinate Systems

Coordinate systems form the framework for locating points in space and understanding their relationships with each other. In three dimensions, we use a 3D coordinate system, often called the Cartesian coordinate system.

• A point's position in 3D space is determined by three coordinates: \((x, y, z)\).
• The x-coordinate shows the position along the horizontal axis.
• The y-coordinate is the position along the vertical depth.
• The z-coordinate indicates the height in the space.

For example, a point with coordinates \((0, 0, z)\) lies on the z-axis. If both x and y values are zero, you are neither moving sideways (x-direction) nor forward or backward (y-direction) but moving exclusively along the z. Being familiar with the coordinate systems helps in setting up and solving problems involving lines, shapes, and volumes in three dimensions.

Lines in 3D Space

Lines in 3D space are fundamental elements in geometry and calculus, representing the pathways between points in three-dimensional environments. Unlike lines in 2D, which are defined by a linear equation, lines in 3D often need parametric equations for accurate representation.A line in 3D can be defined using a point on the line and a direction vector that indicates the line's orientation. Parametric equations show each coordinate \((x, y, z)\) as functions of a parameter \(t\). For example, given a point \((x_0, y_0, z_0)\) and direction vector \(\langle a, b, c \rangle\), the parametric form is:

• \( x(t) = x_0 + at \)
• \( y(t) = y_0 + bt \)
• \( z(t) = z_0 + ct \)

These equations allow the calculation of any point along the line as \( t \) varies. This method helps simplify the representation of complex path trajectories and can describe many geometric scenarios in engineering, architecture, and physics, supporting robust spatial analysis.