Problem 17 - 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. $$(0,1,1), \quad(0,-1,1)$$

Step-by-Step Solution

Answer

Parametrization: \(x(t) = 0\), \(y(t) = 1-2t\), \(z(t) = 1\); for \(0 \leq t \leq 1\).

Step 1: Identify Points and Direction

The line segment joins two points \(A = (0,1,1)\) and \(B = (0,-1,1)\). The direction vector \(\vec{v}\) for the line segment is given by \((x_2 - x_1, y_2 - y_1, z_2 - z_1)\). In this case, \((0-0, -1-1, 1-1) = (0, -2, 0)\). Hence, the direction vector \(\vec{v} = (0, -2, 0)\).

Step 2: Write Parametrization

The parametrization of a line segment from point \(A\) to \(B\) is given by \(\vec{r}(t) = \vec{A} + t\vec{v}\), where \(0 \leq t \leq 1\). For our line segment: \((x(t), y(t), z(t)) = (0, 1, 1) + t(0, -2, 0) = (0, 1-2t, 1)\).

Step 3: Interpret Parametrization

The parametrization \(x(t) = 0\), \(y(t) = 1-2t\), \(z(t) = 1\) defines the line segment between the given points. As \(t\) increases from 0 to 1, the point moves from \(A\) to \(B\).

Step 4: Sketch the Parametrization

Draw a 3D coordinate system and plot the points \(A = (0,1,1)\) and \(B = (0,-1,1)\). The line segment is a straight line parallel to the y-axis in the plane \(x=0\) and \(z=1\). The direction of increasing \(t\) is from \(A\) to \(B\). Indicate this with an arrow on the line segment from (0,1,1) to (0,-1,1).

Key Concepts

Direction Vector

The direction vector is a fundamental component in understanding line segments in three-dimensional space. It gives us information about the orientation and direction of the line segment. The direction vector is typically computed by subtracting the coordinates of the initial point from the terminal point. In the given problem, the line segment is defined between points \( A = (0, 1, 1) \) and \( B = (0, -1, 1) \). Here’s how to calculate it step by step:

• Subtract the x-coordinates: \( 0 - 0 = 0 \)
• Subtract the y-coordinates: \( -1 - 1 = -2 \)
• Subtract the z-coordinates: \( 1 - 1 = 0 \)

Thus, the direction vector \( \vec{v} \) is \( (0, -2, 0) \). This vector indicates that our line segment moves vertically, without any change along the x or z axes. This highlights that the line segment is parallel to the y-axis.

3D Coordinate System

Understanding the 3D coordinate system is essential to visualize and comprehend the orientation of objects and points in space. In a 3-Dimensional space, each point is represented by a set of coordinates (x, y, z), indicating its position along the x, y, and z axes respectively. For this problem:

• The point \(A = (0, 1, 1)\) is located at x = 0, y = 1, z = 1.
• The point \(B = (0, -1, 1)\) is at x = 0, y = -1, z = 1.

Notice that both points lie on the plane where x=0 and z=1, which implies that our segment is constrained within this plane.
In this scenario, drawing the axes can help:

• Picture the Y-axis as the vertical line increasing upwards and downwards.
• The Z-axis extends outward towards you or back behind the screen.
• The X-axis goes from left to right.

The line segment from A to B on a 3D plane will look parallel to the Y-axis since their direction vector \( \vec{v} = (0, -2, 0)\) indicates that solely the y-coordinate changes.

Parametric Equations

Parametric equations are a powerful tool to represent lines, curves, and surfaces using parameters, often denoted as \( t \). They provide a flexible way to express the position of points over a segment or curve based on this parameter. For a line segment, parametric equations help trace the path between two points as the parameter \( t \) varies between 0 and 1.To find the parametrization of a line segment connecting points \( A \) and \( B \), we use the formula:\[ \vec{r}(t) = \vec{A} + t\vec{v} \] where \( \vec{A} \) is the starting point, \( \vec{v} \) is the direction vector, and \( t \) is the parameter.In this case:

• The starting point A is \((0, 1, 1)\).
• The direction vector \( \vec{v} \) is \((0, -2, 0)\).
• Thus, the parametric equations become \( x(t) = 0 \), \( y(t) = 1 - 2t \), \( z(t) = 1 \).

As \( t \) progresses from 0 to 1, the point \( (x(t), y(t), z(t)) \) moves smoothly from \( A \) to \( B \) along the line segment. This setup graphically showcases how a point traverses through space over time, governed by the direction vector and the initial point.