Problem 199 - Calculus Volume 3

Question

Suppose that the position function for an object in three dimensions is given by the equation \(\mathbf{r}(t)=t \cos (t) \mathbf{i}+t \sin (t) \mathbf{j}+3 t \mathbf{k}\) Show that the particle moves on a circular cone.

Step-by-Step Solution

Answer

The particle moves on a circular cone with the axis along the z-direction.

Step 1: Understanding the Position Vector

The position function \(\mathbf{r}(t) = t \cos(t) \mathbf{i} + t \sin(t) \mathbf{j} + 3t \mathbf{k}\) represents the trajectory of a particle in three-dimensional space. Here \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) are the unit vectors along the x, y, and z axes, respectively.

Step 2: Expressing Parametric Equations

The position vector components give us parametric equations: \(x = t \cos(t)\), \(y = t \sin(t)\), and \(z = 3t\). These equations describe how each coordinate changes with respect to the parameter \(t\).

Step 3: Finding the Relationship Between x and y

From the parametric equations for \(x\) and \(y\), use the identity \(\cos^2(t) + \sin^2(t) = 1\) to find \(x^2 + y^2 = t^2(\cos^2(t) + \sin^2(t)) = t^2\). This simplifies to \(x^2 + y^2 = t^2\).

Step 4: Relating x, y, and z to Confirm the Cone

We now have \(x^2 + y^2 = t^2\) and \(z = 3t\). Expressing \(t\) from the z-equation gives \(t = \frac{z}{3}\). Substitute this back into \(x^2 + y^2 = t^2\), yielding \(x^2 + y^2 = \left(\frac{z}{3}\right)^2\). This is the equation of a cone with an axis along the z-direction.

Key Concepts

Position Vector

A position vector is a fundamental concept in multivariable calculus, representing the location of a point in space relative to a given origin. In our problem, the position function is given as \( \mathbf{r}(t) = t \cos(t) \mathbf{i} + t \sin(t) \mathbf{j} + 3t \mathbf{k} \), which describes the path of a particle in three-dimensional space. Here, \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) are unit vectors along the x, y, and z axes, respectively.

The position vector \( \mathbf{r}(t) \) has three components:

• \( x(t) = t \cos(t) \)
• \( y(t) = t \sin(t) \)
• \( z(t) = 3t \)

These components specify the location of the particle at any time \( t \). Each term of the vector gives the particle's coordinate in its respective axis in a dynamic way, depending on the parameter \( t \).

Understanding the position vector helps to visualize the trajectory and behavior of particles or objects moving in three-dimensional space.

Parametric Equations

In the study of multivariable calculus, parametric equations play a crucial role in describing paths and curves in space. For the given problem, the position vector \( \mathbf{r}(t) \) provides us with the parametric equations: \( x = t \cos(t) \), \( y = t \sin(t) \), and \( z = 3t \).

Parametric equations express the coordinates of the points that make up a curve or path as functions of a single parameter, here denoted as \( t \). This allows us to capture complex trajectories and shapes that would be difficult to express using standard functions.

By manipulating the parameter \( t \), one can determine the position of the particle at any specific moment, effectively mapping out its trajectory in a 3D space.

These equations are particularly useful because they allow us to separate each axis's behavior, making it easier to study the relationships and interactions between them.

Circular Cone

A circular cone is a type of three-dimensional geometric shape characterized by a circular base and a single vertex, with lines (generators) extending from this base to the vertex. In mathematical terms, the problem shows that the path traced by the particle creates such a cone.

To confirm that the particle moves on a cone, the solution involves analyzing the parametric equations. We find \( x^2 + y^2 = t^2 \) using the trigonometric identity \( \cos^2(t) + \sin^2(t) = 1 \). This simplifies our understanding as it indicates a circular cross-section.

Further examination with \( z = 3t \) helps us relate the vertical component, demonstrating the linear relationship between \( t \) and \( z \). Ultimately, solving \( x^2 + y^2 = \left(\frac{z}{3}\right)^2 \) confirms it forms a cone, verifying the three-dimensional curve of the path the particle follows.

3D Trajectory

Understanding the 3D trajectory of a particle is crucial for visualizing its movement through space. In the current exercise, the trajectory is defined by the position vector \( \mathbf{r}(t) = t \cos(t) \mathbf{i} + t \sin(t) \mathbf{j} + 3t \mathbf{k} \).

By examining the parametric equations, we can follow the particle's path as it moves over time. The motion is not only limited to a plane but extends into three dimensions.

The trajectory of the particle can be best visualized by thinking of how the path expands as \( t \) increases, spiraling around the z-axis and forming a shape often referred to as a helix or a helical path. However, the given solution shows us it traces the surface of a cone, a unique motion.

Recognizing 3D trajectories allows for better understanding in physics and engineering, where paths need to be analyzed and predicted accurately.