Problem 9 - Calculus Early Transcendentals

Question

\(7-14\) Sketch the curve with the given vector equation. Indicate with an arrow the direction in which \(t\) increases. $$ \mathbf{r}(t)=\langle t, \cos 2 t, \sin 2 t\rangle $$

Step-by-Step Solution

Answer

The curve is a helix around the x-axis with direction increasing as t increases.

Step 1: Understanding the Vector Equation

The given vector equation is \( \mathbf{r}(t) = \langle t, \cos 2t, \sin 2t \rangle \). This describes a three-dimensional curve where \( t \) represents the \( x \)-coordinate, \( \cos 2t \) represents the \( y \)-coordinate, and \( \sin 2t \) represents the \( z \)-coordinate.

Step 2: Identifying the Structure of the Curve

We can observe from the vector equation that as \( t \) varies, the \( y \) and \( z \) components \( (\cos 2t, \sin 2t) \) trace out a circle in the \( y-z \) plane because \( \cos^2 2t + \sin^2 2t = 1 \). The \( x \)-coordinate increases linearly with \( t \), describing a helical path around the \( x \)-axis.

Step 3: Plotting Points for Key Values of \( t \)

Plot points for various values of \( t \), such as \( t=0, \pi/2, \pi, 3\pi/2, 2\pi \), to understand the curve. At \( t=0 \), the point is \( \langle 0, 1, 0 \rangle \); at \( t=\pi/2 \), the point is \( \langle \pi/2, 0, 1 \rangle \); at \( t=\pi \), it is \( \langle \pi, -1, 0 \rangle \); at \( t=3\pi/2 \), it is \( \langle 3\pi/2, 0, -1 \rangle \); and at \( t=2\pi \), it is back to \( \langle 2\pi, 1, 0 \rangle \).

Step 4: Indicating the Direction of the Curve

The direction of the curve as \( t \) increases is along the path of the helix wrapping around the \( x \)-axis. As \( t \) increases, we move from one point to the next plotted point in order of increasing \( t \). Arrows are drawn along the helical path to show this increasing direction.

Step 5: Sketching the Helix

With the points plotted and the direction indicated, sketch the curve. Start by drawing a helix where each loop corresponds to one full revolution of the circle in the \( y-z \) plane, moving forward along the \( x \)-axis. Use the arrows to indicate the increasing direction of the parameter \( t \).

Key Concepts

Helical Curve

A helical curve, also known as a helix, is a fascinating three-dimensional shape that resembles a spiraling spring or coil. To picture a helix, imagine a slinky toy stretched out along a horizontal rod. A helix is characterized by its shape as it winds around an axis. In our vector equation example, the helix wraps around the x-axis, which serves as the central spine. This type of helical structure is typical in nature and technology, seen in DNA strands and spiral staircases.

The uniqueness of a helix lies in its continuous, smooth curve that forms as it progresses along the axis. This suggests that as you move along the x-axis, each corresponding position on the path traces a circle in the yz-plane. The helical path described by our vector equation can be seen as a translation of this circular motion along a new dimension, extending it into a three-dimensional space. Such curves find numerous applications in mechanical devices and architectural structures.

3D Sketching

3D sketching is a valuable skill that helps us visualize and understand three-dimensional objects more effectively. Sketching the helical curve from our vector exercise requires us to think about how the curve moves through space. To sketch this, start by visualizing the core structure of the helix, which is built upon the circular pattern in the yz-plane. This circle is continuously translated along the x-axis as the parameter t changes.

When sketching in 3D, it is helpful to use a coordinate system to determine points on the curve. As seen in the solution:

• At t=0, the point is located at ⟨0, 1, 0⟩.
• At t=π/2, the point moves to ⟨π/2, 0, 1⟩.
• At t=π, it shifts to ⟨π, -1, 0⟩, continuing this pattern.

By plotting these points, connect them using a smooth curve to depict the helical spiral winding around the x-axis. Importantly, arrows should be added to indicate the upward direction along the x-axis as t increases, helping to convey the curve's dynamic motion through three dimensions.

Parametric Equations

Parametric equations are a powerful mathematical tool that let us describe curves in multiple dimensions using parameters, usually denoted by t. These equations specify each coordinate of a point based on a single parameter, offering a clear view into complex curves.

The vector equation from our exercise, \( \mathbf{r}(t) = \langle t, \cos 2t, \sin 2t \rangle \), serves as an excellent example of how parametric equations work in three dimensions. Here:

• The x-coordinate is determined by the parameter t, fluctuating as t changes.
• The y-coordinate follows \( \cos 2t \), indicating periodic behavior that creates the rotating aspect around the x-axis.
• Similarly, the z-coordinate's \( \sin 2t \) contributes to the circular motion within the yz-plane.

This approach allows us to model complex and realistic forms, such as the helical pattern, with precision. Using parametric equations is fundamental in fields such as computer graphics, engineering, and physics, where it is essential to describe and manipulate three-dimensional forms accurately.