Problem 113 - Precalculus with Limits

Question

GRAPHICAL REASONING Consider two forces \(\mathbf{F}_1 = \langle 10, 0 \rangle\) and \(\mathbf{F}_2 = 5\langle \cos\ \theta, \sin\ \theta \rangle\). (a) Find ||\(\mathbf{F}_1 + \mathbf{F}_2\)|| as a function of \(\theta\). (b) Use a graphing utility to graph the function in part (a) for \(0 \leq \theta < 2\pi\). (c) Use the graph in part (b) to determine the range of the function. What is its maximum, and for what value of \(\theta\) does it occur? What is its minimum, and for what value of \(\theta\) does it occur? (d) Explain why the magnitude of the resultant is never 0.

Step-by-Step Solution

Answer

The function ||\(\mathbf{F}||\) is \(\sqrt{(10+5\cos\theta)^2 + (5\sin\theta)^2}\). Its graph has a maximum of 15 at \(\theta = \pi\) and a minimum of 10 at \(\theta = 0\). Its magnitude can never be zero because the x-component of \(\mathbf{F}\) has a constant term that is not zero.

Step 1: Combine the Vectors

First combine the vectors \(\mathbf{F}_1\) and \(\mathbf{F}_2\). To do this, add together corresponding components of the vectors to obtain a resultant vector \(\mathbf{F} = \mathbf{F}_1 + \mathbf{F}_2\). In this case, \(\mathbf{F}\) will be \(\langle 10+5\cos\theta, 5\sin\theta \rangle\).

Step 2: Find the Magnitude of the Resultant Vector

The magnitude of a vector \(\mathbf{v} = \langle a, b \rangle\) in a 2D space is \(\sqrt{a^2 + b^2}\). Applying this formula to \(\mathbf{F}\), we obtain ||\(\mathbf{F}|| = \(\sqrt{(10+5\cos\theta)^2 + (5\sin\theta)^2}\). This is the function of \(\theta\) we were asked to find in part (a).

Step 3: Graph the Function

Use a graphing utility to plot the function ||\(\mathbf{F}||\) for the specified range of \(\theta\). The function is periodic with a period of \(2\pi\).

Step 4: Analyze the Graph

Inspect the graph to find the maximum and minimum values of ||\(\mathbf{F}||\). Notice that the maximum occurs at \(\theta = \pi\) and the minimum at \(\theta = 0\), and these extremes are 15 and 10 respectively. This is the range of the function.

Step 5: Explain Why the Magnitude of the Resultant Is Never 0

The magnitude of the resultant vector \(\mathbf{F}\) will be zero only if both its components are zero. But the x-component (10+5cos\(\theta\)) can never be zero due to the constant term 10. Hence, the magnitude of \(\mathbf{F}\) can never be zero.

Key Concepts

Magnitude of a Vector

Understanding the magnitude of a vector is key in solving many physics and mathematics problems. The magnitude of a vector is essentially the length of the vector, and it provides a measure of how much force, speed, or other quantity the vector represents. In a two-dimensional space, for any vector \(\mathbf{v} = \langle a, b \rangle\), the formula for the magnitude \(||\mathbf{v}||\) is given by:

• \( ||\mathbf{v}|| = \sqrt{a^2 + b^2} \)

This formula is derived from applying the Pythagorean theorem to the vector's components plotted on a coordinate plane. You'll often need to calculate this when combining vectors, as this allows us to quantify the net effect. When given a resultant vector, such as \(\mathbf{F} = \langle 10+5\cos\theta, 5\sin\theta \rangle\), finding its magnitude lets us understand the overall strength of the combined forces.

Graphing Functions

Graphing functions is a powerful visual tool that helps us understand and analyze behaviors of mathematical expressions. By using a graphing utility, you can visually represent the relationship between variables, like \(\theta\) in functions of the form \(||\mathbf{F}||\).
To graph a function, first identify the range of your variable, here \(0 \leq \theta < 2\pi\). The function, based on the magnitude of the resultant vector, lets you observe how this value changes as \(\theta\) varies.

• Plot the x-axis with values from 0 to \(2\pi\) and the y-axis representing the magnitude function \(||\mathbf{F}||\).
• Use software or a calculator capable of graphing trigonometric functions.

This plot will show the periodic nature of the function. Look for peaks and troughs to identify maximum and minimum values as these often relate directly to real-world maximum and minimum phenomena.

Resultant Vector

The concept of a resultant vector arises when you add two or more vectors together. It's the single vector that has the same effect as the individual vectors combined. In physical contexts, it might represent the total force or displacement.
To find the resultant vector of two vectors, like \(\mathbf{F}_1 = \langle 10, 0 \rangle\) and \(\mathbf{F}_2 = 5\langle \cos\theta, \sin\theta \rangle\), you add their components separately:

• Resultant x-component: \(10 + 5\cos\theta\)
• Resultant y-component: \(5\sin\theta\)

Together, they form the vector \(\mathbf{F} = \langle 10+5\cos\theta, 5\sin\theta \rangle\), which you then analyze to understand the total effect—both direction and magnitude. This vector is essential in many engineering and physics problems involving forces or motion.

Trigonometric Functions

Trigonometric functions are integral in relating the angles and sides of triangles, but they extend far beyond pure geometry. In vectors, functions like \(\cos\theta\) and \(\sin\theta\) often represent direction components, particularly in polar coordinates or rotational systems.
The trigonometric expressions in the vector \(\mathbf{F}_2 = 5\langle \cos\theta, \sin\theta \rangle\) show how the direction changes with \(\theta\).

• \(\cos\theta\) reflects how much of the vector points along the x-axis.
• \(\sin\theta\) shows the component along the y-axis.

These functions are periodic, with a period of \(2\pi\), meaning they repeat their values in this interval. Grasping these can help you predict and understand motion in oscillating systems, waves, or even in complex electrical circuits.