Problem 41 - University Calculus: Early Transcendentals

Question

Describe the given set with a single equation or with a pair of equations. The line through the point (1,3,-1) parallel to the a. \(x\) -axis b. \(y\) -axis c. \(z\) -axis

Step-by-Step Solution

Answer

x-axis: \(x = 1+t, y = 3, z = -1\); y-axis: \(x = 1, y = 3+t, z = -1\); z-axis: \(x = 1, y = 3, z = -1+t\).

Step 1: Identify Direction for the x-axis

The line is parallel to the x-axis, which means its direction vector points along the x direction. Since it passes through the point (1,3,-1), we can represent the direction vector as (1,0,0), indicating movement only along the x-axis.

Step 2: Formulate Equation for x-axis Parallel Line

The parametric equations for a line passing through a point \((x_0, y_0, z_0)\) with direction vector \((a, b, c)\) are \(x = x_0 + at\), \(y = y_0 + bt\), \(z = z_0 + ct\). For the x-axis, these become: \(x = 1 + t\), \(y = 3\), \(z = -1\).

Step 3: Identify Direction for the y-axis

The line is parallel to the y-axis, meaning its direction vector is along the y-axis. Thus, the direction vector is \((0, 1, 0)\) because the line has no x or z component.

Step 4: Formulate Equation for y-axis Parallel Line

Similarly, using the point-direction form equation with the point \((1, 3, -1)\) and direction \((0, 1, 0)\), the parametric equations become: \(x = 1\), \(y = 3 + t\), \(z = -1\).

Step 5: Identify Direction for the z-axis

The line parallel to the z-axis has a direction vector \((0, 0, 1)\). This indicates movement along the z-axis only.

Step 6: Formulate Equation for z-axis Parallel Line

Using the point \((1, 3, -1)\) and direction vector \((0, 0, 1)\), the parametric equations are: \(x = 1\), \(y = 3\), \(z = -1 + t\).

Key Concepts

Direction Vector

In the context of parametric equations and lines in 3D space, understanding the concept of a direction vector is crucial. A direction vector indicates the direction in which a line extends. For example, when a line is said to be parallel to the x-axis, its direction vector is

• (1, 0, 0)

This means that the line moves along the x-axis, and its y and z components remain unchanged.
Similarly, if a line is parallel to the y-axis, the direction vector is

• (0, 1, 0)

indicating movement in the y-direction while remaining stationary along x and z. When dealing with z-axis parallel lines, the direction vector becomes

• (0, 0, 1)

meaning the line only moves along the z-axis.
Direction vectors are an essential tool in understanding how lines behave in 3D space.

Point-Direction Form

The point-direction form is a method used to represent lines in three-dimensional geometry. This form allows us to describe a line by specifying a point through which the line passes and a vector that indicates its direction. The general parametric equation of a line using the point-direction form is:

• For a line passing through point oes: (x_0, y_0, z_0) with direction vector *(a, b, c):*
  • x = x_0 + at
  • y = y_0 + bt
  • z = z_0 + ct

Here, *t* represents a parameter that can take any real value, essentially "sliding" along the line. By plugging different values of *t* into these equations, you can find various points on the line. This form is powerful because it very clearly ties geometry (the location of the line) with algebra (the equations we use).

Axis Parallel Lines

Axis parallel lines are a core concept when dealing with 3D coordinate geometry. These lines either follow the path of the x-axis, y-axis, or z-axis with no deviation into other directions. Let's break this down with examples:

• If a line is parallel to the x-axis, its parametric form can simplify to: x = x_0 + t
  y = y_0
  z = z_0


• When parallel to the y-axis: x = x_0
  y = y_0 + t
  z = z_0


• And if parallel to the z-axis: x = x_0
  y = y_0
  z = z_0 + t

This simplification occurs because the movement only occurs along one axis, leaving the others constants.

3D Coordinate Geometry

3D coordinate geometry extends the concepts from two-dimensional geometry into three dimensions, allowing us to analyze and describe the spatial relationships between points, lines, and planes. In this versatile mathematical framework:

• Each point is determined by three coordinates (x, y, z), indicating its location relative to the three perpendicular axes.
• You can use parametric equations to describe lines in this space, providing insights into their orientation and positioning.
• By coupling point-direction forms and direction vectors, we can handle complex problems involving parallel, intersecting, or skew lines.

3D geometry finds applications in various fields such as computer graphics, engineering, and physics. Understanding its basic principles, like creating equations for axis-parallel lines, sets a foundation for more advanced studies.