Problem 13 - University Calculus: Early Transcendentals

Question

Find parametrizations for the line segments joining the points. Draw coordinate axes and sketch each segment, indicating the direction of increasing \(t\) for your parametrization. $$(0,0,0), \quad(1,1,3 / 2)$$

Step-by-Step Solution

Answer

Parametrize with \(x(t) = t\), \(y(t) = t\), \(z(t) = \frac{3}{2}t\) for \(0 \leq t \leq 1\).

Step 1: Identify Initial and Final Points

The line segment starts at point \(A = (0, 0, 0)\) and ends at point \(B = (1, 1, 3/2)\).

Step 2: Determine the Direction Vector

The direction vector \(\mathbf{v}\) is found by subtracting the coordinates of the initial point from the final point. Thus, \(\mathbf{v} = (1-0, 1-0, 3/2-0) = (1, 1, 3/2)\).

Step 3: Write the Parametric Equations

The parametric equations for the line segment can be written as:\[ x(t) = 0 + 1t = t \]\[ y(t) = 0 + 1t = t \]\[ z(t) = 0 + \frac{3}{2}t = \frac{3}{2}t \] where \(0 \leq t \leq 1\).

Step 4: Sketch the Segment with Parameter Direction

Draw a 3D coordinate system and plot the points \(A(0,0,0)\) and \(B(1,1,3/2)\). Draw a line segment between these points. Indicate an arrow on the segment pointing from \(A\) to \(B\) to show the direction of increasing \(t\).

Key Concepts

Line Segment Parametrization

Understanding the concept of line segment parametrization is key to describing how a line travels in space. In mathematics, a line segment is defined by two specific points: a starting point and an endpoint. Parametrization allows us to express this segment using a set of equations that depend on a parameter, often denoted as \(t\).

This parameter \(t\) usually varies from 0 to 1, helping us determine a specific point on the line segment for every value of \(t\). For the segment from \((0, 0, 0)\) to \((1, 1, 3/2)\), the equations describing the parametrization are:

• \(x(t) = t\)
• \(y(t) = t\)
• \(z(t) = \frac{3}{2} t\)

What this means is:

• At \(t = 0\), we are at the starting point \((0, 0, 0)\)
• At \(t = 1\), we reach the endpoint \((1, 1, 3/2)\)

This method provides a simple way to visualize and understand the position of any point along the segment at any given time.

Direction Vector

The direction vector is an essential concept when working with parametric equations to define a path or line segment in space. It tells us the direction in which the line progresses from one point to another.

To find the direction vector, you subtract the coordinates of the initial point from the final point. Given our points \(A = (0, 0, 0)\) and \(B = (1, 1, 3/2)\), the direction vector \(\mathbf{v}\) is calculated as:

\[ \mathbf{v} = (1 - 0, 1 - 0, \frac{3}{2} - 0) = (1, 1, \frac{3}{2}) \]

This vector is critical because it determines how changes in the parameter \(t\) translate to movements in the 3D space. Essentially, it scales the movement such that as \(t\) moves from 0 to 1, a complete transition from the starting point to the endpoint occurs.

An intuitive way to think about the direction vector is as a set of instructions for traveling from the initial point to the terminal point along the line segment.

3D Coordinate System

A 3D coordinate system is a framework that helps visualize and locate points or objects in three-dimensional space. It consists of three axes, usually labeled \(x\), \(y\), and \(z\), which intersect at a point called the origin \((0, 0, 0)\).

In a 3D space, each point can be represented by a triple of numbers \((x, y, z)\), corresponding to its distances along the \(x\)-axis, \(y\)-axis, and \(z\)-axis, respectively. For example, in our exercise:

• The point \(A = (0, 0, 0)\) is the origin, where all axes intersect.
• The point \(B = (1, 1, 3/2)\) is one unit along the \(x\) and \(y\) axes and 1.5 units along the \(z\) axis.

When sketching the line segment between these points in a 3D coordinate system, it's crucial to accurately place each point and draw the segment between them. To maintain accuracy:

• Ensure the axes are drawn perpendicular to one another.
• Use consistent scaling to ensure a proportional representation.

This setup helps in visualizing how the parametric equations describe the path of the line segment in three-dimensional space, providing a clearer understanding of the spatial relationships involved.