Problem 15 - Essential Calculus Early Transcendentals

Question

Find parametric equations for the line segment from \((10,3,1)\) to \((5,6,-3) .\)

Step-by-Step Solution

Answer

The parametric equations are \(x(t) = 10 - 5t\), \(y(t) = 3 + 3t\), \(z(t) = 1 - 4t\) for \(0 \leq t \leq 1\).

Step 1: Identify the Points

The line segment is defined from point \((10,3,1)\) to point \((5,6,-3)\). Let's denote these points as \(A = (10, 3, 1)\) and \(B = (5, 6, -3)\).

Step 2: Determine the Direction Vector

The direction vector \( \mathbf{d} \) can be found by subtracting the coordinates of point \(A\) from point \(B\). So, \( \mathbf{d} = (B_x - A_x, B_y - A_y, B_z - A_z) = (5 - 10, 6 - 3, -3 - 1) = (-5, 3, -4)\).

Step 3: Define the Parametric Equations

The parametric equations for the line segment are given by \((x, y, z) = A + t \mathbf{d}\), where \(t\) ranges from 0 to 1. The equations are:\[ x(t) = 10 + t(-5) = 10 - 5t \] \[ y(t) = 3 + t(3) = 3 + 3t \] \[ z(t) = 1 + t(-4) = 1 - 4t \] for \( 0 \leq t \leq 1 \).

Step 4: Confirm the End Points

By setting \(t = 0\), you obtain \(x = 10\), \(y = 3\), and \(z = 1\), which is the starting point. By setting \(t = 1\), you obtain \(x = 5\), \(y = 6\), and \(z = -3\), which is the ending point, verifying our parameterization.

Key Concepts

Direction Vector

Understanding the concept of a direction vector is crucial for solving problems in 3D geometry, such as finding parametric equations for a line segment. A direction vector provides the path from one point to another. In our exercise, we need to determine a line segment from \( (10, 3, 1) \) to \((5, 6, -3)\). The direction vector \(\mathbf{d}\) for the line is computed by taking the difference of corresponding coordinates from the second point to the first point:

• Subtract the x-coordinates: \(5 - 10 = -5\)
• Subtract the y-coordinates: \(6 - 3 = 3\)
• Subtract the z-coordinates: \(-3 - 1 = -4\)

Thus, the direction vector is \(\mathbf{d} = (-5, 3, -4)\). This vector tells us in which direction to move from the starting point \((10, 3, 1)\) to reach the endpoint \((5, 6, -3)\). It's like having a compass that guides us along a straight path in a 3D space.

Line Segment

A line segment is part of a line that has two defined endpoints, in contrast to a full line which extends infinitely in both directions. When dealing with line segments in parametric equations, it's important to establish the initial and terminal points clearly.
In our example, the line segment connects point \(A = (10, 3, 1)\) to point \(B = (5, 6, -3)\). The parametric representation uses a parameter \(t\) to express how far along the direction vector the line segment extends:

• When \(t = 0\), the position is \(A\), the starting point.
• When \(t = 1\), the position is \(B\), the ending point.

As \(t\) transitions from 0 to 1, it traces out the segment from \(A\) to \(B\), ensuring we stay within the boundary of the line segment.

3D Geometry

Involving three dimensions — length, width, and height — 3D geometry deals with objects that have volume. Parametric equations in 3D geometry help express where an object lies within this space and how it is oriented.
In our task, we convert the line segment into parametric form, which reveals its formation in a 3D space. The parametric equations derived were:

• \(x(t) = 10 - 5t\)
• \(y(t) = 3 + 3t\)
• \(z(t) = 1 - 4t\)

These equations depict how each coordinate changes as we progress along the line segment from point A to point B. It's an elegant way to describe motion and shape using algebra and vectors in a three-dimensional world.

Mathematical Reasoning

Using mathematical reasoning, we logically deduce and verify facts and relationships. It involves recognizing patterns, making conjectures, and forming arguments to support conclusions.
In this exercise, we use reasoning to ensure that our parametric representation starts and ends at the correct points. By setting \(t = 0\), we check that we indeed start at \((10, 3, 1)\). Similarly, by setting \(t = 1\), we confirm the endpoint \((5, 6, -3)\) is reached.
Additionally, reasoning about the direction vector ensures that our parameterization is accurate. This involves understanding vector operations and the properties of parametric equations. Our analysis and verification show the correctness of the mathematics used to solve the problem, demonstrating how calculation and logic work together.