Problem 7 - Essential Calculus Early Transcendentals

Question

\(5-12=\) Sketch the curve with the given vector equation. Indicate with an arrow the direction in which \(t\) increases. $$ \mathbf{r}(t)=\langle t, 2-t, 2 t\rangle $$

Step-by-Step Solution

Answer

Sketch the line decreasing in y, increasing in x and z, starting at (0, 2, 0) with an arrow pointing towards increasing t.

Step 1: Understanding the Vector Function

The given vector equation \( \mathbf{r}(t)=\langle t, 2-t, 2t \rangle \) represents a parametric curve in 3-dimensional space, with \( t \) acting as the parameter. The coordinates \( (x, y, z) \) at any point on the curve are given by \( x = t \), \( y = 2 - t \), and \( z = 2t \).

Step 2: Analyzing the Component Functions

To understand the shape and direction of the curve, consider each component of \(\mathbf{r}(t)\). As \( t \) varies, \( x = t \) suggests that the x-coordinate increases linearly. Similarly, \( z = 2t \) indicates that z also increases linearly, but at twice the rate of x. \( y = 2 - t \) decreases linearly as \( t \) increases.

Step 3: Sketching the Curve

Begin by drawing the coordinate axes. For \( t = 0 \), the curve starts at (0, 2, 0). As \( t \) increases, plot additional points like (1, 1, 2), (2, 0, 4), and (3, -1, 6). Connect these points smoothly to form a line. The path will descend along the y-axis while increasing in both x and z-directions.

Step 4: Indicating the Direction of Increase

To show the direction in which \( t \) increases, draw an arrow along the curve. The arrow should point from (0, 2, 0) in the direction of (1, 1, 2), indicating increasing values of \( t \) generally move the curve into more negative y-values and more positive x and z-values.

Key Concepts

Parametric Equations

Parametric equations are a way to express a set of quantities as explicit functions of a single parameter, often represented as \( t \). Instead of describing a curve by a single equation, we use a set of equations to describe each coordinate separately:

• \( x = f(t) \)
• \( y = g(t) \)
• \( z = h(t) \)

These equations are particularly useful in describing curves in spaces of higher dimensions, like 3D. In a parametric equation, the parameter often represents time or progression along the curve. This gives the benefit of providing a complete description of the curve’s behavior and movement.

In the provided vector function \( \mathbf{r}(t) = \langle t, 2-t, 2t \rangle \), we understand how each component evolves as \( t \) changes. Here, the parameter \( t \) directs how far along the curve we travel, guiding how the points move through space with respect to time or progression.

3D Coordinate Systems

In three-dimensional space, positions are described using three coordinates: \( (x, y, z) \). These correspond to distances along three perpendicular axes, generally referred to as the x-axis, y-axis, and z-axis.

• The x-axis typically lies in a horizontal plane.
• The y-axis usually runs vertically.
• The z-axis extends out of the plane formed by the x and y axes, creating depth.

Understanding 3D Coordinate Systems is crucial when dealing with vector functions like the one in this exercise. Here, the vector \( \mathbf{r}(t)=\langle t, 2-t, 2t \rangle \) demonstrates how a point traces out a curve in 3D space as \( t \) changes. Each component function—\( x(t) = t \), \( y(t) = 2-t \), and \( z(t) = 2t \)—highlights how the point's position depends on \( t \).

Being comfortable with 3D coordinate systems helps you visualize how these components work together to form a path or trace a curve through the 3D space.

Vector Curves

A vector curve is a mathematical representation of a curve in space, defined by a vector function of one or more variables. In the context of our original exercise, the vector curve is represented by \( \mathbf{r}(t)=\langle t, 2-t, 2t \rangle \). This means, for any particular \( t \), you can find a specific point on the curve. The point is a combination of three components:

• The x-component: \( t \)
• The y-component: \( 2-t \)
• The z-component: \( 2t \)

Visualizing this can help understand the behavior of the curve. By observing how each component changes as \( t \) increases, you notice that the x and z components increase while the y component decreases. This results in the curve moving through the 3D space and gives it a directional flow.
An arrow is typically drawn along the curve to show the direction of increasing \( t \). This directional aspect is crucial in physics and engineering, where understanding not just the path but the direction and speed along it forms a significant part of analysis. Knowing how vector curves behave is essential for fields that model movement and dynamics in three dimensions.