Problem 13 - Calculus Early Transcendentals

Question

Find the distance between the points with polar coordinates \((2, \pi / 3)\) and \((4,2 \pi / 3) .\)

Step-by-Step Solution

Answer

The distance is \(2\sqrt{3}\).

Step 1: Convert Polar Coordinates to Rectangular Coordinates

To find the distance between two points with polar coordinates, it's easier to convert them to rectangular coordinates (Cartesian coordinates) first. For a point \((r, \theta)\), the rectangular coordinates are \((x,y)\), where \(x = r \cdot \cos(\theta)\) and \(y = r \cdot \sin(\theta) \).First point \((2, \pi/3)\):\[ x_1 = 2 \cdot \cos(\pi/3) = 2 \cdot \frac{1}{2} = 1 \]\[ y_1 = 2 \cdot \sin(\pi/3) = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3} \]Second point \((4, 2\pi/3)\):\[ x_2 = 4 \cdot \cos(2\pi/3) = 4 \cdot \left(-\frac{1}{2}\right) = -2 \]\[ y_2 = 4 \cdot \sin(2\pi/3) = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3} \]The rectangular coordinates are \((x_1, y_1) = (1, \sqrt{3})\) and \((x_2, y_2) = (-2, 2\sqrt{3})\).

Step 2: Apply the Distance Formula

The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) in the Cartesian plane is calculated using the distance formula:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Substitute the coordinates:\[ x_1 = 1, \quad y_1 = \sqrt{3}, \quad x_2 = -2, \quad y_2 = 2\sqrt{3} \]\[ d = \sqrt{(-2 - 1)^2 + (2\sqrt{3} - \sqrt{3})^2} \]\[ d = \sqrt{(-3)^2 + (\sqrt{3})^2} \]\[ d = \sqrt{9 + 3} \]\[ d = \sqrt{12} \]\[ d = 2\sqrt{3} \]

Step 3: Conclusion: Interpret the Result

The calculation shows that the distance between the points given in polar coordinates \((2, \pi/3)\) and \((4, 2\pi/3)\) is \(2\sqrt{3}\) units on the plane. Since distances are always positive measures, this result makes sense.

Key Concepts

Understanding Polar Coordinates

Polar coordinates are a way of locating a point in a plane using a distance and an angle. This system represents points with two pieces of information: the radius, usually denoted as \( r \), and the angle \( \theta \) from the positive x-axis.

The pair \((r, \theta)\) describes the point such that:

• \( r \) is the distance from the origin to the point.
• \( \theta \) is the angle measured in radians from the positive x-axis.

For example, a point \((2, \pi/3)\) means a point that is 2 units away from the origin, in the direction of an angle \( \pi/3 \) radians. This means that polar coordinates are often used when dealing with curves and rotations as they directly describe circular motion.

Unlike rectangular coordinates, which describe a point by its horizontal and vertical positions, polar coordinates gain relevance in problems where the relationship to a central point is crucial, like in trigonometry and geometry involving circles.

• It simplifies expressions involving multi-looped figures.
• Makes rotation and spirals calculations more straightforward.

Rectangular Coordinates Explained

Rectangular coordinates, also known as Cartesian coordinates, are the more familiar method of expressing the location of a point in a graph or coordinate system. In this system, each point is represented by an \((x, y)\) pair.

This is how the components work:

• \( x \): The horizontal distance from the origin (0,0).
• \( y \): The vertical distance from the origin (0,0).

The transformation from polar to rectangular is done using trigonometric functions. Given \((r, \theta)\), we calculate:

• \( x = r \cdot \cos(\theta) \)
• \( y = r \cdot \sin(\theta) \)

These conversions are essential when we want to apply formulas like the distance formula, which are easier with rectangular coordinates. In our exercise, we transformed \((2, \pi/3)\) and \((4, 2\pi/3)\) into \((1, \sqrt{3})\) and \((-2, 2\sqrt{3})\) to simplify calculations.

Understanding this relationship helps in transitioning between circular references in polar format to linear mapping in rectangular systems, thereby expanding our ability to tackle various geometric problems.

Using the Distance Formula

The distance formula is a mathematical equation used to determine the straight-line distance between two points in a Cartesian coordinate system. Once converted to rectangular coordinates, determining the distance between any two points follows a consistent pattern.

Suppose we have two points \((x_1, y_1)\) and \((x_2, y_2)\). The distance \(d\) is calculated as:

• \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

This formula is rooted in the Pythagorean theorem, considering the differences in coordinates to form a right triangle.

In our example, the points \((1, \sqrt{3})\) and \((-2, 2\sqrt{3})\) were used to calculate the distance:

• First, calculate differences \((x_2 - x_1) = -2 - 1 = -3\) and \((y_2 - y_1) = 2\sqrt{3} - \sqrt{3} = \sqrt{3}\).
• Square those differences and sum them up: \((-3)^2 + (\sqrt{3})^2 = 9 + 3 = 12\).
• Finally, take the square root to find \(d = \sqrt{12} = 2\sqrt{3}\).

This method not only helps in verifying calculations related to geometry involving polar coordinates but also facilitates in understanding real-world distances objectively.