Problem 16 - 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. $$(1,1,0), \quad(1,1,1)$$

Step-by-Step Solution

Answer

The parametrization for the line segment is \(\mathbf{r}(t) = (1,1,t)\).

Step 1: Understand the Problem

We have two points in 3D space: \((1,1,0)\) and \((1,1,1)\). The task is to find a parameterization that describes the line segment connecting these two points and to illustrate the segment on a graph, including directionality based on an increasing parameter \(t\).

Step 2: Determine the Parametrization

The general formula for parameterizing a line segment between two points \(A(x_1, y_1, z_1)\) and \(B(x_2, y_2, z_2)\) is given by the vector equation: \(\mathbf{r}(t) = (1-t)A + tB\), where \(0 \leq t \leq 1\). Here, \(A = (1,1,0)\) and \(B = (1,1,1)\).

Step 3: Apply the Parametrization Formula

Substitute the coordinates of points \(A\) and \(B\) into the generalized vector equation: \[\mathbf{r}(t) = (1-t)(1,1,0) + t(1,1,1)\]. Simplify the expression: \[\mathbf{r}(t) = (1,1,t)\]. This equation represents the line segment as \(t\) varies from 0 to 1.

Step 4: Illustrate the Line Segment

Visualize the line segment by plotting the points on a 3D graph with axes labeled \(x\), \(y\), and \(z\). Begin at the point \((1,1,0)\) when \(t=0\) and move towards \((1,1,1)\) as \(t\) approaches 1. Use an arrow to indicate the direction of increasing \(t\).

Step 5: Conclusion

The parametrization \(\mathbf{r}(t) = (1,1,t)\) successfully describes the line segment joining the two points. The segment is vertical along the \(z\)-axis, with the \(x\) and \(y\) coordinates constant.

Key Concepts

3D Space

Three-dimensional (3D) space is an advanced concept where any point is represented by three coordinates:

• The x-coordinate, which determines the position along the horizontal axis,
• The y-coordinate, which determines the position along the vertical axis in the plane, and
• The z-coordinate, which adds depth and determines the position in the space above and below the origin plane.

Imagine a room where you can move left-right (x), forward-backward (y), and up-down (z). Each point inside has a unique combination of (x, y, z) coordinates that precisely locates it in 3D space. To fully understand 3D space, visualize it as a cube or a box where each corner or any point inside can be touched using this coordinate system.
Thinking in 3D allows us to address complex geometrical problems involving volume and space representation. In exercises involving line segments, the ability to move through 3D space is essential to understanding how these segments behave and where they lead.

Line Segment

A line segment is simply the shortest path connecting two points. In 3D space, a line segment has some special characteristics:

• It is characterized by its endpoints, in this case, \((1,1,0)\) and \((1,1,1)\).
• It has a definite length, unlike lines that stretch infinitely. The length can be thought of as how far you would walk along it.
• Line segments in 3D show how to transition from one point in space to another.

In our example, the line segment moves from the start point at the origin of the 3D plot, extending only along the z-axis due to constant x and y coordinates. It is a good way to understand spatial relationships by observing line segments and how they define paths and lengths connecting distinct points.

Coordinate Axes

The coordinate axes in 3D space serve as a frame of reference for locating points and shapes. They are:

• The x-axis runs horizontally and indicates movement left or right.
• The y-axis runs vertically, signifying upward or downward movement on the plane.
• The z-axis goes in and out of the plane, representing depth.

In exercises involving line segments, like the one we're analyzing, coordinate axes help create a visual framework. Drawing axes helps us see directionality and motion. In our instance:

• As you move from the point \((1,1,0)\) to \((1,1,1)\), the travel is purely along the z-axis. The x and y values remain unchanged.

This insight is crucial to understanding how each axis behaves and affects movement in 3D space.

Parametrization Formula

A parametrization formula is a vital tool for representing line segments in mathematics. For a line segment between two points \(A(x_1, y_1, z_1)\) and \(B(x_2, y_2, z_2)\), the parameterization is typically given by \[\mathbf{r}(t) = (1-t)A + tB\].
Here's what happens with the formula:

• The variable \(t\) is a parameter which ranges between 0 and 1.
• When \(t=0\), the equation returns the starting point, \(A\).
• When \(t=1\), it lands at the ending point, \(B\).
• Values of \(t\) between 0 and 1 trace the line segment between \(A\) and \(B\).

This technique is invaluable for visualizing and tracing paths smoothly. In our solution, the formula \(\mathbf{r}(t) = (1,1,t)\) transitions between the 3D points by making x and y constant, while z changes with \(t\). This provides a clear understanding of how the line segment is constructed as \(t\) varies.