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Chapter 12: Problem 12

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

Short Answer

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

Step by step solution

01

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.
02

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.
03

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: \( z = t \).
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥攍et's say (a, b, c)鈥攊ndicate 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鈥攕pecifically 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.

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Most popular questions from this chapter

Which of the following are always true, and which are not always true? Give reasons for your answers. \(\begin{array}{ll}{\text { a. } \mathbf{u} \cdot \mathbf{v}=\mathbf{v} \cdot \mathbf{u}} & {\text { b. } \mathbf{u} \times \mathbf{v}=-(\mathbf{v} \times \mathbf{u})} \\ {\mathbf{c} \cdot(-\mathbf{u}) \times \mathbf{v}=-(\mathbf{u} \times \mathbf{v})} & {}\end{array}\) d. \((c \mathbf{u}) \cdot \mathbf{v}=\mathbf{u} \cdot(c \mathbf{v})=c(\mathbf{u} \cdot \mathbf{v}) \quad\) (any number \(c )\) e. \(c(\mathbf{u} \times \mathbf{v})=(c \mathbf{u}) \times \mathbf{v}=\mathbf{u} \times(c \mathbf{v}) \quad\) (any number \(c )\) \(\mathbf{f} . \mathbf{u} \cdot \mathbf{u}=|\mathbf{u}|^{2} \quad\) g. \((\mathbf{u} \times \mathbf{u}) \cdot \mathbf{u}=0\) \(\mathbf{h} .(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{u}=\mathbf{v} \cdot(\mathbf{u} \times \mathbf{v})\)

Show that the volume of the segment cut from the paraboloid $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=\frac{z}{c} $$ by the plane \(z=h\) equals half the segment's base times its altitude. (Figure 12.49 shows the segment for the special case \(h=c . )\)

Find the areas of the parallelograms whose vertices are given in Exercises \(35-38 .\) $$ A(1,0), \quad B(0,1), \quad C(-1,0), \quad D(0,-1) $$

Sketch the surfaces in Exercises \(13-76\) $$ \left(y^{2} / 4\right)-\left(x^{2} / 4\right)-z^{2}=1 $$

Sketch the surfaces in Exercises \(13-76\) $$ y^{2}-x^{2}=z $$
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