Problem 11 - 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. $$ \mathbf{r}(t)=t^{2} \mathbf{i}+t^{4} \mathbf{j}+t^{6} \mathbf{k} $$

Step-by-Step Solution

Answer

Sketch a 3D curve starting at the origin, with arrows indicating increase along positive axes as \( t \) increases.

Step 1: Understanding the Vector Equation

The given vector equation is \( \mathbf{r}(t) = t^2 \mathbf{i} + t^4 \mathbf{j} + t^6 \mathbf{k} \). This means that the position vector \( \mathbf{r}(t) \) at any time \( t \) is defined by the components \( t^2 \) along the \( x \)-axis, \( t^4 \) along the \( y \)-axis, and \( t^6 \) along the \( z \)-axis.

Step 2: Analyzing the Components

Let's look at the individual components to see how they behave as \( t \) increases:- The \( x \)-component \( t^2 \) is non-negative and increases quadratically. - The \( y \)-component \( t^4 \) also increases, but more steeply (quartically).- The \( z \)-component \( t^6 \) increases the most steeply (sextically).Since all components are powers of \( t \), they all increase as \( t \) increases.

Step 3: Sketching the Curve

Begin by plotting key points of the curve for specific values of \( t \), such as \( t = 0, 1, -1, 2, -2 \). Notice the trend where, as \( t \) moves away from zero, the vector components get significantly larger, especially in the \( z \)-direction due to the \( t^6 \) term. The curve results in a 3D path that starts at the origin when \( t = 0 \) and spirals outwards.

Step 4: Indicating the Direction

Draw arrows along the curve to indicate the direction in which \( t \) increases. From the analysis, we know that the vector components increase as \( t \) becomes positive. For negative \( t \), components are still positive but smaller (since even powers of negatives are positive), and the path retraces to approach the origin.

Key Concepts

Vector Equations

Vector equations are used to describe geometric locations or paths as functions of a variable, typically time (denoted by \( t \)). In this case, the vector equation \( \mathbf{r}(t) = t^2 \mathbf{i} + t^4 \mathbf{j} + t^6 \mathbf{k} \) specifies a 3D path. It breaks down into components associated with the \( x \)-, \( y \)-, and \( z \)-directions, represented by \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) respectively.
Understanding vector equations help in visualizing trajectories and motions in space, often employed in physics and engineering.
When tackling vector equations, it's vital to:

• Identify each component vector and observe how they influence the overall position vector.
• Understand how changes in the parameter \( t \) affect each component.

This makes it easier to predict the shape and direction of the path they describe.

3D Coordinate Systems

The three-dimensional coordinate system is fundamental in visualizing and analyzing problems in space. It consists of three axes:

• The \( x \)-axis pointing horizontally.
• The \( y \)-axis often representing vertical motion, though by convention in math it's often longitudinal.
• The \( z \)-axis, which adds depth by extending perpendicularly to the \( xy \)-plane.

Each point in 3D space is defined by its position relative to these axes, expressed as \((x, y, z)\).
Understanding 3D coordinate systems enables us to precisely locate objects and visualize vectors like \( \mathbf{r}(t) = t^2 \mathbf{i} + t^4 \mathbf{j} + t^6 \mathbf{k} \), interpreting their geometric meanings as physical paths or areas.
Such systems are crucial for modeling real-world phenomena, from flight dynamics to architecture.

Vector Components

Vector components break down vectors into parts that align with axis directions, such as \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) in 3D space. These components represent how much of the vector lies along each axis.
In the vector equation \( \mathbf{r}(t) = t^2 \mathbf{i} + t^4 \mathbf{j} + t^6 \mathbf{k} \):

• \(t^2 \mathbf{i}\) indicates the movement along the \( x \)-axis, growing quadratically as \( t \) increases.
• \(t^4 \mathbf{j}\) shows the growth along the \( y \)-axis, in a quartic manner.
• \(t^6 \mathbf{k}\) represents steep growth along the \( z \)-axis, sextically.

Understanding vector components helps visualize how vectors change in different spatial directions, empowering problem-solving in fields ranging from graphic design to mechanical engineering.