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Chapter 12: Problem 28

Sketch the graph of r(t) and show the direction of increasing t. $$ \mathbf{r}(t)=9 \cos t \mathbf{i}+4 \sin t \mathbf{j}+t \mathbf{k} $$

Short Answer

Expert verified
The graph is a 3D helix with elliptical cross-sections, moving upwards as t increases.

Step by step solution

01

Identify the Components of the Vector Function

The vector function is given as \( \mathbf{r}(t)=9 \cos t \mathbf{i}+4 \sin t \mathbf{j}+t \mathbf{k} \). It consists of three components: \( x(t) = 9 \cos(t) \), \( y(t) = 4 \sin(t) \), and \( z(t) = t \).
02

Determine the Shape of the Projection in the XY-Plane

The expressions \( x(t) = 9 \cos(t) \) and \( y(t) = 4 \sin(t) \) define an ellipse. More generally, \( x^2/81 + y^2/16 = 1 \), which describes an ellipse centered at the origin with semi-major axis 9 along the x-axis and semi-minor axis 4 along the y-axis.
03

Understand the Parametric Representation in 3D

The vector function can be visualized in 3D as a helix. This is because the \( z \)-component, \( z(t) = t \), increases linearly with \( t \), causing the ellipse described in the \( xy \)-plane to shift upwards as \( t \) increases. This gives the overall path of a helix.
04

Sketch the Helix in 3D

Draw the base ellipse in the \( xy \)-plane. Then, draw lines showing the ellipse moving upwards as \( t \) increases, forming a helical shape. Each complete cycle in the \( xy \)-plane corresponds to a 2蟺 increase in \( z \), resulting in a full loop of the helix.
05

Indicate the Direction of Increasing \( t \)

Starting from a point where \( t = 0 \), the vector \( \mathbf{r}(t) \) moves in a counterclockwise direction when viewed from above the \( xy \)-plane, as this represents the increase in \( t \) in the parametric equations. Mark arrows along the helix to show this direction of increasing \( t \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Helix
A helix is a fascinating geometric shape that arises naturally in the curly strands of DNA or telephone cords. In mathematics, a helix is a type of space curve with a constant radius and consistent twisting. In our exercise, the vector function \( \mathbf{r}(t)=9 \cos t \mathbf{i}+4 \sin t \mathbf{j}+t \mathbf{k} \) elegantly illustrates a helix.
  • The function's first two components, \(9 \cos t\) and \(4 \sin t\), describe the elliptical base in the \(xy\)-plane.
  • The third component, \(z(t) = t\), introduces a linear change along the z-axis, causing the ellipse to elevate and form a helix.
By combining circular (elliptical in this case) motion with a constant rise, the helix continuously spirals upwards. When \( t\) is viewed increasing positively, our helix moves counterclockwise, a detail that represents time's natural progression. This demonstration across Cartesian coordinates brings to life the intertwined essence of 3D space.
3D Parametric Plots
3D parametric plots allow us to represent complex forms, like our helix, comprehensively. By defining functions for each axis, we reveal a spatial story with intriguing depth and dimension.
  • In the given function, \(x(t) = 9 \cos(t)\), \(y(t) = 4 \sin(t)\), and \(z(t) = t\), each component contributes meaningfully to the plot's shape.
  • While \(x(t)\) and \(y(t)\) position the point within the horizontal plane, \(z(t)\) determines how the point ascends.
Understanding these plots nurtures an appreciation for how vector functions map out paths that defy two-dimensional limitations. These graphs not only showcase mathematical beauty but also mimic many natural phenomena and engineering designs, bridging abstract theory and real-world applications.
Ellipse
At the heart of the vector function's representation lies an ellipse, a smooth, closed curve that generalizes the concept of a circle.
  • For our function, \(x(t) = 9 \cos(t)\) and \(y(t) = 4 \sin(t)\) trace an ellipse鈥檚 path in the \(xy\)-plane.
  • The equation \( \frac{x^2}{81} + \frac{y^2}{16} = 1 \) highlights its elliptical nature, with its center at the origin.
  • The semi-major axis of 9 aligns with the x-axis, and the semi-minor axis of 4 aligns with the y-axis, reinforcing the stretched circle appearance that defines ellipses.
Ellipses are instrumental in several advanced fields, such as astronomy, where they describe planetary orbits. Understanding their formation through parametric equations equips us with a versatile toolset for analyzing analogous situations in various scientific contexts.

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Most popular questions from this chapter

Find the displacement and the distance traveled over the indicated time interval. $$ \mathbf{r}=t^{2} \mathbf{i}+\frac{1}{3} t^{3} \mathbf{j} ; \quad 1 \leq t \leq 3 $$

Assume that \(s\) is an arc length parameter for a smooth vector-valued function \(\mathbf{r}(s)\) in 3 -space and that \(d \mathbf{T} / d s\) and \(d \mathbf{N} / d s\) exist at each point on the curve. (This implies that \(d \mathbf{B} / d s\) exists as well, since \(\mathbf{B}=\mathbf{T} \times \mathbf{N} .)\) The following derivatives, known as the Frenet-Serret formulas, are fundamental in the theory of curves in 3 -space: $$\begin{array}{ll}{d \mathbf{T} / d s=\kappa \mathbf{N}} & {[\text { Exercise } 58]} \\ {d \mathbf{N} / d s=-\kappa \mathbf{T}+\tau \mathbf{B}} & {[\text { Exercise } 60]} \\ {d \mathbf{B} / d s=-\tau \mathbf{N}} & {[\text { Exercise } 59(c)]}\end{array}$$ Use the first two Frenet-Serret formulas and the fact that \(\mathbf{r}^{\prime}(s)=\mathbf{T}\) if \(\mathbf{r}=\mathbf{r}(s)\) to show that $$ \tau=\frac{\left[\mathbf{r}^{\prime}(s) \times \mathbf{r}^{\prime \prime}(s)\right] \cdot \mathbf{r}^{\prime \prime \prime}(s)}{\left\|\mathbf{r}^{\prime \prime}(s)\right\|^{2}} \quad \text { and } \quad \mathbf{B}=\frac{\mathbf{r}^{\prime}(s) \times \mathbf{r}^{\prime \prime}(s)}{\left\|\mathbf{r}^{\prime \prime}(s)\right\|} $$

Use the given information to find the position and velocity vectors of the particle. $$ \mathbf{a}(t)=\sin t \mathbf{i}+\cos t \mathbf{j}+e^{t} \mathbf{k} ; \mathbf{v}(0)=\mathbf{k} ; \mathbf{r}(0)=-\mathbf{i}+\mathbf{k} $$

Suppose that the position function of a particle moving in 3 -space is \(\mathbf{r}=3 \cos 2 t \mathbf{i}+\sin 2 t \mathbf{j}+4 t \mathbf{k} .\) (a) Use a graphing utility to graph the speed of the particle versus time from \(t=0\) to \(t=\pi\) (b) Use the graph to estimate the maximum and minimum speeds of the particle. (c) Use the graph to estimate the time at which the maximum speed first occurs. (d) Find the exact values of the maximum and minimum speeds and the exact time at which the maximum speed first occurs.

Show that the angle between \(\mathbf{v}\) and \(\mathbf{a}\) is constant for the position vector \(\mathbf{r}=e^{t} \cos t \mathbf{i}+e^{t} \sin t \mathbf{j} .\) Find the angle.
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