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 parametrization is \(\mathbf{r}(t) = (t,0,0)\) for \(t\in[0,1]\).

Step 1: Understanding the Problem

We are tasked with finding a parametrization for a line segment joining two points in 3D space: \((0,0,0)\) and \((1,0,0)\). A parametrization is an expression that describes a vector line using a parameter \(t\) that varies over a specified interval.

Step 2: Define Parametric Equations

The parametric equations for a line segment can be written as \( \mathbf{r}(t) = (1-t) \mathbf{a} + t\mathbf{b} \), where \( \mathbf{a} \) and \( \mathbf{b} \) are the endpoints of the segment. Here, \( \mathbf{a} = (0,0,0) \) and \( \mathbf{b} = (1,0,0) \).

Step 3: Substitute the Endpoints

Substitute \( \mathbf{a} \) and \( \mathbf{b} \) into the parametric equation: \[ \mathbf{r}(t) = (1-t)(0,0,0) + t(1,0,0) = (t,0,0) \].

Step 4: Determine the Interval for t

For a line segment, the parameter \(t\) typically ranges from 0 to 1. As \(t\) varies from 0 to 1, the point \((t,0,0)\) moves from \((0,0,0)\) to \((1,0,0)\).

Step 5: Sketch the Line Segment

Draw a 3D coordinate system. Mark the points \((0,0,0)\) and \((1,0,0)\) on the x-axis. Sketch a straight line segment between these two points. Indicate the direction of increasing \(t\) from \((0,0,0)\) to \((1,0,0)\).

Key Concepts

3D Coordinate System

A 3D coordinate system allows us to navigate through three dimensions: length, width, and height. This is essential for representing objects and understanding their positions in space.
Each point in 3D space can be described using coordinates \(x, y, z\), which specify a location along the x, y, and z axes. The zero point, or origin, is \(0, 0, 0\).
When solving problems in a 3D coordinate system, it's crucial to visualize how the axes interact. The x-axis typically runs horizontally, the y-axis vertically, and the z-axis perpendicularly to establish depth.

• X-Axis: Represents horizontal direction.
• Y-Axis: Represents vertical direction.
• Z-Axis: Introduces the third dimension for depth.

Understanding this helps when sketching lines or segments between points, as you can effectively trace how changes along each axis affect the location of points in space.

Line Segment Parametrization

Parametrization is a powerful technique in mathematics, especially when dealing with line segments in 3D. It allows you to express the line in terms of a variable, often called a parameter \t\, which smoothly transitions along the segment.
For a line segment joining two points, say \( \mathbf{a} = (x_1, y_1, z_1)\) and \( \mathbf{b} = (x_2, y_2, z_2)\), the parametric equation is:\[\mathbf{r}(t) = (1-t) \mathbf{a} + t\mathbf{b}\]This equation is versatile because as \( t \) changes from 0 to 1, the expression moves continuously from point \mathbf{a} to point \mathbf{b\). The formula effectively divides the segment into parts based on \( t \) and captures every point along the segment.
Breaking down the equation:

• The factor \( (1-t)\) scales the starting point, ensuring that when \( t = 0 \, you start at \mathbf{a}\.
• The factor \( t\) scales the ending point, allowing a smooth transition until \( t = 1 \), reaching \mathbf{b}\.

This method simplifies complex 3D problems by transforming them into more manageable and readable parametric expressions.

Vector Mathematics

Vector mathematics underpins the analysis and computation involving 3D space. A vector represents both magnitude and direction, important in distinguishing how objects move or are positioned in 3D.
Vectors are typically denoted with a bold letter or as a list in parentheses like \( \mathbf{v} = (v_x, v_y, v_z)\). They can describe a point's displacement from the origin or between two points' difference.

• Addition: Summing vectors combines their respective components: \( \mathbf{u} + \mathbf{v} = (u_x+v_x, u_y+v_y, u_z+v_z)\).
• Scalar Multiplication: Scaling a vector by a number alters its magnitude without changing its direction (unless the scalar is negative, which also reverses the vector): \( c\mathbf{v} = (cv_x, cv_y, cv_z)\).

In problem-solving, vectors allow for straightforward manipulation and representation of lines and surfaces in space. They simplify computations in parametrization, as seen with the line segment, offering a clear approach to navigate through and visualize 3D problems.