Problem 38 - Thomas Calculus

Question

Show that the vector-valued function \(\mathbf{r}(t)=(2 \mathbf{i}+2 \mathbf{j}+\mathbf{k})\) $$ +\cos t\left(\frac{1}{\sqrt{2}} \mathbf{i}-\frac{1}{\sqrt{2}} \mathbf{j}\right)+\sin t\left(\frac{1}{\sqrt{3}} \mathbf{i}+\frac{1}{\sqrt{3}} \mathbf{j}+\frac{1}{\sqrt{3}} \mathbf{k}\right) $$ describes the motion of a particle moving in the circle of radius 1 centered at the point \((2,2,1)\) and lying in the plane \(x+y-2 z=2\) .

Step-by-Step Solution

Answer

The function describes a circle of radius 1 centered at (2,2,1) on the plane x+y-2z=2.

Step 1: Understand the Problem

We need to verify that the vector-valued function \( \mathbf{r}(t) \) represents a circle of radius 1 centered at \((2, 2, 1)\) and lying in the plane \( x + y - 2z = 2 \). This involves checking both the radius and the plane conditions.

Step 2: Analyze the Center and Radius

Examine the components \((2 \mathbf{i} + 2 \mathbf{j} + \mathbf{k})\) as the center of the circle. The radius is determined by analyzing the coefficients of the trigonometric functions. Notice that the terms \( \frac{1}{\sqrt{2}} \) and \( \frac{1}{\sqrt{3}} \) are unit-length and orthogonal, ensuring that the trigonometric terms form a circle of radius 1.

Step 3: Verify the Plane Condition

Substitute \( \mathbf{r}(t) = (x(t), y(t), z(t)) \) into the plane equation \( x + y - 2z = 2 \). Given the structure of \( \mathbf{r}(t) \), simplify to ensure the equation holds for all \( t \). After simplification, \( (2 + \frac{1}{\sqrt{2}}\cos t + \frac{1}{\sqrt{3}}\sin t) + (2 - \frac{1}{\sqrt{2}} \cos t + \frac{1}{\sqrt{3}}\sin t) - 2(1 + \frac{1}{\sqrt{3}}\sin t) = 2 \) holds, confirming the plane.

Key Concepts

vector-valued function

A vector-valued function is essentially a function where each output is a vector. In the context of mechanics and physics, it's frequently used to describe the trajectory of moving objects. Here, the vector-valued function \( \mathbf{r}(t) \) represents the position of a particle at any time \( t \). This function is a combination of vectors, with each component defining the position in the \( x \), \( y \), and \( z \) directions. Understanding vector-valued functions is crucial for visualizing how an object moves through three-dimensional space over time. In \( \mathbf{r}(t) = (2 \mathbf{i} + 2 \mathbf{j} + \mathbf{k}) + \cos t \left( \frac{1}{\sqrt{2}} \mathbf{i} - \frac{1}{\sqrt{2}} \mathbf{j} \right) + \sin t \left( \frac{1}{\sqrt{3}} \mathbf{i} + \frac{1}{\sqrt{3}} \mathbf{j} + \frac{1}{\sqrt{3}} \mathbf{k} \right) \), the terms inside form transformations that affect the particle's path, helping visualize its circular movement within a specified plane.

parametric equations

Parametric equations are a way to express a set of related quantities as explicit functions of one or more independent variables known as parameters. In vector calculus, these equations are often used to describe curves in the plane or in space. For our vector-valued function \( \mathbf{r}(t) \), the components \( (x(t), y(t), z(t)) \) can be seen as parametric equations for the particle's path. Each component is a function of \( t \), allowing for a clear understanding of how the particle's position changes over time. So if you need to understand the layout of a path or a geometric figure, parametric equations become an indispensable tool. In this case, the trigonometric components \( \cos t \) and \( \sin t \) create circular motions, intricately linking these motions back to traditional trigonometric concepts.

plane geometry

Plane geometry deals with shapes on a flat surface, like lines, circles, and polygons within a plane. In this exercise, we consider a plane defined by the equation \( x + y - 2z = 2 \). The vector-valued function \( \mathbf{r}(t) \) describes motion confined to this plane. By substituting \( x(t), y(t), \) and \( z(t) \) from \( \mathbf{r}(t) \) into this equation, we verify that the particle’s path lies on the plane. Plane geometry becomes instrumental here as it helps us understand constraints on movement imposed by having to stay on a planar surface in three-dimensional space. Recognizing that all these points satisfy the plane condition solidifies the role of plane geometry in combining abstract equations with tangible spatial concepts.

trigonometry

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles, but it also deals more broadly with periodic functions such as sine and cosine. These functions are crucial for analyzing oscillatory behavior in various sciences, particularly in calculating trajectory paths in vector calculus. In the vector-valued function \( \mathbf{r}(t) \), trigonometric functions \( \cos t \) and \( \sin t \) are used to ensure that the motion is circular, providing both the periodic nature and directionality of the path. Each component that includes these functions contributes to the overall shape and direction of the path described by \( \mathbf{r}(t) \) as it rotates about the defined center, producing a smooth and continuous movement. Understanding trigonometry, therefore, is critical for deciphering how these vector expressions translate into actual movement.