Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the parametric equations of the line through \((5,-3,4)\) that intersects the \(z\)-axis at a right angle.

Step-by-Step Solution

Answer

Parametric equations are \( x = 5 + t, y = -3 + t, z = 4 \).

Step 1: Understand the Problem

We need to find parametric equations for a line passing through the point \((5, -3, 4)\) and intersects the \(z\)-axis at right angles. A line that intersects the \(z\)-axis at a right angle means it is horizontal and doesn't change in \(z\) as it moves through space.

Step 2: Determine Direction Vector

Since the line intersects the \(z\)-axis at a right angle, it lies in the \(xy\)-plane. Thus, its direction vector needs no \(k\) component. Let the direction vector be \( \mathbf{v} = (a, b, 0) \), where \(a\) and \(b\) are any non-zero values that determine the line's direction in the \(xy\)-plane.

Step 3: Write Parametric Equations

The parametric equations of a line are generally given by \((x, y, z) = (x_0, y_0, z_0) + t(a, b, c)\), where \((x_0, y_0, z_0)\) is a point on the line, \( (a, b, c) \) is the direction vector, and \(t\) is the parameter. Here, our point is \((5, -3, 4)\), and the direction vector is \((a, b, 0)\). Thus, the parametric equations are: \[ x = 5 + at \] \[ y = -3 + bt \] \[ z = 4 \]

Step 4: Simplify in Terms of Independent Variables

Since the direction vector can be any vector in the \(xy\)-plane, select convenient values like \((1, 0, 0)\) or \((0, 1, 0)\) for ease. Here, let's choose \( (1, 1, 0) \) for variety: \( a = 1 \), \( b = 1 \) gives: \[ x = 5 + t \] \[ y = -3 + t \] \[ z = 4 \]

Key Concepts

Z-Axis Intersection

When a line intersects the z-axis, it means it crosses or touches it in some manner. However, in this case, the line intersects at a right angle, implying it does not move up or down along the z-axis. Instead, it only moves horizontally in the x-y plane. This means that no matter how far you extend the line, the z-coordinate remains constant.
This concept is crucial when working with 3D geometry, as it helps visually simplify scenarios by reducing the problem space to two dimensions (the x-y plane). Understanding how a line can interact with the z-axis without having a vertical component allows us to visualize paths or trajectories that are inherently horizontal. In practical terms, identifying such intersections can help pinpoint crucial boundary and constraint conditions in physics and engineering applications.

Direction Vector

A direction vector determines the orientation or path that a line takes through space. It is disconnected from any specific point, focusing solely on how the line extends in 3D space. For the line intersecting at a right angle with the z-axis, we only need to consider its projection onto the x-y plane.
Here, the direction vector is expressed as \( \mathbf{v} = (a, b, 0) \), where the '0' indicates that there is no movement in the z-direction. The variables \(a\) and \(b\) are chosen to show how the line extends within the x-y plane, creating a two-dimensional path.

• \(a\) can determine horizontal movement.
• \(b\) can dictate vertical actions in this plane.

In this situation, you can select any non-zero values for \(a\) and \(b\), with common choices being \((1, 0, 0)\), \((0, 1, 0)\), or honestly, any combination that aids the problem at hand. The selection simplifies calculations and offers flexibility in describing paths or trajectories in horizontal planes.

Parametric Form

Parametric equations are a versatile method for expressing a line or curve using one or more parameters. For lines in 3D geometry, these equations elegantly capture the line's trajectory, specifying x, y, and z coordinates as functions of a single variable. Typically, this variable \(t\) acts as a parameter that drives the point along the line over time.
The general form is shown by \((x, y, z) = (x_0, y_0, z_0) + t(a, b, c)\), where:

• \((x_0, y_0, z_0)\) denotes a known point on the line.
• \((a, b, c)\) represents the direction vector.

In our case, since the line passes through \((5, -3, 4)\) with a direction vector of \((a, b, 0)\), the equations simplify to:

• \(x = 5 + at\)
• \(y = -3 + bt\)
• \(z = 4\)

This rearrangement underscores that the z-axis remains unaffected, indicating a horizontal trajectory — perfect for lines situated at a right angle to the z-axis.

3D Geometry

3D geometry involves the study and understanding of spatial forms in three dimensions, where any point can be described using x, y, and z coordinates. This understanding is foundational for many areas, such as physics, engineering, and computer graphics.
In 3D space, lines can travel in any direction, and that direction is defined by vectors. Parametric equations let you precisely quantify these paths, offering a robust way to handle complex spatial worlds. In our scenario, the line's horizontal journey, unaffected by the z-dimension, gives us a clear example of managing constraints in 3D space.

• Visualizing this line can be done by imagining it moving within an invisible horizontal plane parallel to the x-y axes.
• The ability to restrict motion to select dimensions simplifies analysis, making it easier to theorize behaviors and consequences.

Mastering these principles enables students to not only appreciate but also accurately predict interactions and transformations that occur within our rich, three-dimensional environment.