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Chapter 15: Problem 16

Sketch the curve \(C\) determined by \(r(t),\) and indicate the orientation. $$ \mathbf{r}(t)=t \mathbf{i}+2 t \mathbf{j}+e^{t} \mathbf{k} ; \quad t \text { in } \mathbb{R} $$

Short Answer

Expert verified
The curve is a 3D spiral (helix) moving upwards with increasing t.

Step by step solution

01

Analyze the function components

The given vector function is \( \mathbf{r}(t) = t \mathbf{i} + 2t \mathbf{j} + e^{t} \mathbf{k} \). This represents a parametric curve in three dimensions, where the \( x \)-coordinate is \( t \), the \( y \)-coordinate is \( 2t \), and the \( z \)-coordinate is \( e^{t} \).
02

Identify the curve shape in the xy-plane

Projecting \( \mathbf{r}(t) \) onto the xy-plane, we get the function \( (x, y) = (t, 2t) \). This is a line that passes through the origin with a slope of 2.
03

Determine the behavior in the z-direction

The \( z \)-coordinate is \( e^{t} \), an exponential function, which always increases as \( t \) increases. This means that as the curve travels along the xy-line, it simultaneously rises exponentially in the z-direction.
04

Combine components for 3D sketch

The combination of linear motion in the xy-plane and exponential growth in the z-direction results in a spiral or helical curve moving upward. The curve moves counterclockwise when viewed from above the z-axis because \( t \) increases linearly in both the \( x \) and \( y \) directions.
05

Indicate orientation

As \( t \) increases, the orientation of the curve will be along increasing values of \( x, y, \text{ and } z \), forming a curve that spirals upwards.

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Key Concepts

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

Parametric Equations
Parametric equations are a way of describing a set of coordinates in space by expressing them as functions of one or more independent variables, typically symbolized by \( t \).
For example, the vector function \( \mathbf{r}(t) = t \mathbf{i} + 2t \mathbf{j} + e^{t} \mathbf{k} \) describes a path in three-dimensional space.
  • The parameter \( t \) defines the pathway's progression.
  • Each component function (\( x(t), y(t), \) and \( z(t) \)) specifies how the position in space changes as \( t \) varies.
This form is particularly useful for representing curves and surfaces, offering a convenient method to explore geometric properties and behaviors, such as the tangent and velocity vectors.
3D Vector Functions
3D vector functions like \( \mathbf{r}(t) = t \mathbf{i} + 2t \mathbf{j} + e^{t} \mathbf{k} \) describe curves or paths through space using vector components for each spatial dimension.
Here:
  • The \( x \)-component \( t \) indicates linear motion along the x-axis.
  • The \( y \)-component \( 2t \) shows linear motion along the y-axis.
  • The \( z \)-component \( e^{t} \) represents exponential growth along the z-axis.
These components together create a comprehensive picture of how the path behaves in 3D. As \( t \) changes, the function describes not just static positions, but the dynamic flow of a moving curve, unfolding its shape three-dimensionally.
Helical Curves
A helical curve is a type of three-dimensional curve shaped like a helix, often described as a spiral. This is the result of combining distinct types of motion in different dimensions, like in the function \( \mathbf{r}(t) = t \mathbf{i} + 2t \mathbf{j} + e^{t} \mathbf{k} \).
The line component in the \( xy \)-plane (\( t \mathbf{i} + 2t \mathbf{j} \)) indicates straight linear motion, while the \( z \)-component (\( e^{t} \mathbf{k} \)) suggests vertical expansion or rise.
  • The overall effect is a curve that wraps upward, circling around as it ascends.
  • Its appearance resembles a spring or corkscrew when graphed.
This example of helical motion elegantly demonstrates how combining linear and exponential components creates complex yet predictable paths in space.
Curve Orientation
Orientation of a curve involves determining the direction in which the curve is traversed as the parameter \( t \) increases. In our vector function \( \mathbf{r}(t) = t \mathbf{i} + 2t \mathbf{j} + e^{t} \mathbf{k} \), this translates to:
  • Movement in the positive x-direction as \( t \) itself increases.
  • Simultaneous movement in the positive y-direction, doubling the rate of the x-component.
  • An ever-increasing height in the z-direction due to the exponential \( e^{t} \).
The orientation is important for understanding the path's complete tropical motion.
When viewed from above, such a curve appears to spiral counterclockwise around the z-axis, demonstrating the elegant nature of mathematical situational simulations in 3D spaces.

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Most popular questions from this chapter

A point moves along the parabola \(y=x^{2}\) such that the horizontal component of velocity is always 3 . Find the tangential and normal components of acceleration at \(P(1,1)\)

Exer. 1-6: |al Find the unit tangent and normal vectors \(\mathrm{T}(t)\) and \(\mathrm{N}(t)\) for the curve \(C\) determined by \(\mathrm{r}(t)\) (b) Sketch the graph of \(C,\) and show \(\mathrm{T}(t)\) and \(\mathrm{N}(t)\) for the given value of \(t\). $$ \mathbf{r}(t)=t^{3} \mathbf{i}+3 t \mathbf{j} ; \quad t=1 $$

Exer. \(3-4:\) The position of a point moving in a coordinate plane is given by \(\mathbf{r}(t) .\) Find its velocity, acceleration, and speed at time \(t\). $$ \mathbf{r}(t)=(2-\sin t) \mathbf{i}+(1-\cos t) \mathbf{j} $$

If \(r(t)\) is the position vector of a moving point \(P\), find its velocity, acceleration, and speed at the given time \(t\). \(\mathbf{r}(t)=\sqrt{t} \mathbf{i}+(1+\sqrt{t}) \mathbf{j}: \quad t=4\)

Exer. \(23-28\) : Find the points on the given curve at which the curvature is a maximum. $$ y=\sin x $$
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