Problem 14 - Thomas Calculus

Question

Find parametrizations for the line segments joining the points in Exercises \(13-20 .\) Draw coordinate axes and sketch each segment, indicating the direction of increasing \(t\) for your parametrization. $$ (0,0,0), \quad(1,0,0) $$

Step-by-Step Solution

Answer

The segment is parametrized as \((x, y, z) = (t, 0, 0)\) for \(0 \leq t \leq 1\).

Step 1: Understanding Points and Line Segments

The given points are \((0,0,0)\) and \((1,0,0)\). We need to find a parametrization for the line segment joining these two points, using parameter \(t\).

Step 2: Finding the Direction Vector

The direction vector for the line segment is found by subtracting the coordinates of the initial point from the terminal point: \[(1 - 0, 0 - 0, 0 - 0) = (1, 0, 0)\]

Step 3: Writing Parametric Equations

The parametric equations for the line segment using parameter \(t\), which varies from 0 to 1, are derived by scaling the direction vector and adding it to the initial point:\((x, y, z) = (0, 0, 0) + t(1, 0, 0)\)Therefore:\[x = 0 + t \cdot 1 = t\]\[y = 0 + t \cdot 0 = 0\]\[z = 0 + t \cdot 0 = 0\]

Step 4: Parametrization Summary

Thus, the line segment from \((0,0,0)\) to \((1,0,0)\) can be parametrized as:\((x, y, z) = (t, 0, 0)\) where \(0 \leq t \leq 1\).

Step 5: Sketching the Line Segment

To sketch the segment, draw a line from the origin (\(0,0,0\)) to \((1,0,0)\) on the x-axis. The parameter \(t\) starts from 0 at \((0,0,0)\) and increases to 1 at \((1,0,0)\). This indicates the direction of the line segment.

Key Concepts

Line Segments

Line segments are the part of a line that connects two distinct points. These points, often called endpoints, define the boundaries of the segment. Unlike a line, which infinitely extends in both directions, a line segment has a fixed length. In this exercise, the line segment connects the points \(0,0,0\) and \(1,0,0\) on the x-axis.

To understand a line segment better, consider how it differs from a ray or a whole line:

• **Line**: Extends indefinitely in both directions.
• **Ray**: Starts at one point and extends infinitely in one direction.
• **Line Segment**: Connects two endpoints with a definite length.

The distance between these endpoints can be calculated using the distance formula, though for our example, it's simply \(1\), as the points lie on the x-axis. This segment is a great introduction to fundamental geometric principles in three-dimensional space.

Parametric Equations

Parametric equations provide a way of expressing a geometric object, such as a line or curve, using a parameter. This is particularly helpful for line segments, as they let us describe each point on a segment in a flexible and often intuitive way.

For this line segment, we use the parameter \(t\) where \(0 \leq t \leq 1\). As \(t\) moves from 0 to 1, it traces the entire line segment from the starting point to the ending point. The parametric equations are given as:

• For \(x\) coordinate: \(x = t\)
• For \(y\) coordinate: \(y = 0\)
• For \(z\) coordinate: \(z = 0\)

By varying \(t\), we effectively travel from \(0,0,0\) to \(1,0,0\). This kind of representation highlights the role of the parameter, which acts as a slider that governs the progression along the segment.

Coordinate Geometry

Coordinate geometry, often referred to as analytic geometry, blends algebra and geometry to establish a connection between algebraic equations and geometric structures. In the exercise, we utilize this framework to describe geometric objects like points, lines, and segments through algebraic means.

By assigning coordinates to our points, we can clearly visualize their placement in space. In three-dimensional coordinate systems, a point is captured by the x, y, and z values \(\(x,y,z\)\). Coordinate geometry enables straightforward manipulation and analysis of these points. For example:

• The point \(0,0,0\) is at the origin of the coordinate system.
• The point \(1,0,0\) lies one unit away on the x-axis.

This blend of algebraic and geometric concepts makes it easier to solve problems like determining lengths, gradients, and midpoints of segments. It also lays a foundation for more advanced studies in geometry.

Vector Algebra

Vector algebra is used to handle quantities that have both magnitude and direction. In coordinate geometry, vectors prove essential for describing shifts between points and creating consistent frameworks for analyzing spaces.

To find the **direction vector** for our line segment, we subtract the coordinates of the starting point from those of the ending point: \(1-0, 0-0, 0-0 = (1, 0, 0)\). This vector not only represents the direction of our line segment but also its magnitude, or length.

This method offers several benefits:

• It reduces complex spatial concepts into simple mathematical operations.
• It enables easier interpretation through coordinate changes and transformations.
• It provides efficient ways to calculate dot products and cross products.

Knowing how to use vectors and parametric equations permits a deeper comprehension of the spatial layout and precise manipulation of elements within the coordinate space.