Problem 3 - Advanced Engineering Mathematics

Question

Graph the curve traced by the given vector function. $$ \mathbf{r}(t)=t \mathbf{i}+2 t \mathbf{j}+\cos t \mathbf{k} ; t \geq 0 $$

Step-by-Step Solution

Answer

The curve is a helix starting at the origin, moving upwards with oscillation along the z-axis.

Step 1: Identify the Components of the Vector Function

The vector function is given as \( \mathbf{r}(t) = t \mathbf{i} + 2t \mathbf{j} + \cos(t) \mathbf{k} \). This can be broken down into its component functions: \( x(t) = t \), \( y(t) = 2t \), and \( z(t) = \cos(t) \).

Step 2: Analyze the Range of the Parameter

The parameter \( t \) is specified to be greater than or equal to 0, \( t \geq 0 \). This indicates that the graph will commence at \( t = 0 \) and extend indefinitely in the direction defined by the components.

Step 3: Describe the Path in 3D Space

For \( t \geq 0 \), as \( t \) increases, \( x(t) = t \) increases linearly, \( y(t) = 2t \) also increases linearly but twice as fast, and \( z(t) = \cos(t) \) oscillates between -1 and 1 periodically. This suggests the path might be a helical structure climbing as \( t \) increases.

Step 4: Sketch the Graph

To sketch the graph, start with the axis representing the \( x, y, z \) coordinates. For each \( t \), plot the point \( (t, 2t, \cos(t)) \) as \( t \) increases. You'll observe that the points form a helical pattern moving upwards with respect to the \( z \)-axis due to \( \cos(t) \) oscillating, while \( x \) and \( y \) move linearly away from the origin.

Key Concepts

3D graphing

Graphing in three dimensions (3D) involves representing mathematical functions or equations on a set of three axes – typically labeled as the x-axis, y-axis, and z-axis. Unlike 2D graphs, which only use two axes, 3D graphing allows us to visualize how a function behaves over a three-dimensional space.
When creating a 3D graph, each point on the graph corresponds to a set of values \((x, y, z)\) derived from mathematical expressions. These points are marked in relation to the three axes.

• x-axis: This usually represents the first function or parameter and stretches horizontally.
• y-axis: It represents the second function, rising vertically.
• z-axis: This axis typically projects outwards and represents a third expression or parameter.

To graph a 3D function, you must understand the relationship between these three coordinates. In this context, graphing helps us visualize how changes in one or multiple inputs affect the output, presenting a spatial perspective that is crucial for understanding complex mathematical relationships.

parametric equations

Parametric equations are a type of equation used to express mathematical functions in terms of parameters, often represented as \( t \).
Unlike regular equations, which directly express one variable in terms of others (like \( y = f(x) \)), parametric equations use a separate set of equations to define each coordinate in terms of one or more parameters.
This method is particularly useful for tracing curves and complex shapes that can be difficult to describe using standard equations. For instance, in our original exercise:

• \( x(t) = t \)
• \( y(t) = 2t \)
• \( z(t) = \cos(t) \)

Each component function specifies how a point on the curve behaves as the parameter \( t \) varies. As \( t \) changes, these functions provide corresponding \(x, y, z\) coordinates, facilitating the graphing of complex structures in 3D space. With parametric equations, you're essentially 'drawing' a path through space, marking points through the evolution of \( t \).
Understanding parametric equations is crucial because they provide an elegant and powerful way to represent phenomena where independent and dependent variables are intertwined in intricate ways.

helix in 3D space

A helix is a fascinating geometric curve that spirals around a central axis, much like the shape of a spring or a corkscrew. In 3D space, a helix is characterized by its continuous three-dimensional spiral form.
This structure is typically generated when a point moves along a circular path while simultaneously advancing in a linear direction. Such mixed motion results in the spiral climb or descent observed in a helix.
Applying this to our vector function \( \mathbf{r}(t)=t \mathbf{i}+2 t \mathbf{j}+\cos t \mathbf{k} \), we see:

• \(x(t) = t\) and \(y(t) = 2t\) - both functions imply linear movement on the x and y axes.
• \(z(t) = \cos(t)\) - introduces periodic oscillations between -1 and 1.

This combination of components results in a helical pattern, as the \(z\) value rises and falls with \(\cos(t)\), while the \(x\) and \(y\) values steadily increase. This reveals the helix's ascension through 3D space. It reflects the harmony between linear movement and cyclical variation, which is brilliantly captured by parametric equations.
Recognizing helices is essential in various scientific and engineering fields, where such forms appear in objects like DNA strands, springs, and electromagnetic coils.