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Chapter 9: Problem 31

Compute the distance between the given points. (The coordinates are polar coordinates.) $$\left(2, \frac{2 \pi}{3}\right) \text { and }\left(4, \frac{\pi}{6}\right)$$

Short Answer

Expert verified
The distance is \(2\sqrt{5}\).

Step by step solution

01

Convert Polar Coordinates to Cartesian Coordinates

The given points are in polar coordinates: For point 1: - Radius \(r_1 = 2\)- Angle \(\theta_1 = \frac{2\pi}{3}\)For point 2:- Radius \(r_2 = 4\)- Angle \(\theta_2 = \frac{\pi}{6}\)Convert these into Cartesian coordinates using:\[(x, y) = (r \cdot \cos(\theta), r \cdot \sin(\theta))\]So the Cartesian coordinates are:- Point 1: \(x_1 = 2 \cdot \cos\left(\frac{2\pi}{3}\right) = -1\) \(y_1 = 2 \cdot \sin\left(\frac{2\pi}{3}\right) = \sqrt{3}\)- Point 2: \(x_2 = 4 \cdot \cos\left(\frac{\pi}{6}\right) = 2\sqrt{3}\) \(y_2 = 4 \cdot \sin\left(\frac{\pi}{6}\right) = 2\)
02

Apply the Distance Formula

To find the distance between two points in Cartesian coordinates, use the distance formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Substitute the values we calculated in Cartesian coordinates:\[x_1 = -1, \ y_1 = \sqrt{3}\]\[x_2 = 2\sqrt{3}, \ y_2 = 2\]Calculate:\[x_2 - x_1 = 2\sqrt{3} + 1\]\[y_2 - y_1 = 2 - \sqrt{3}\]Substitute into the distance formula:\[d = \sqrt{(2\sqrt{3} + 1)^2 + (2 - \sqrt{3})^2}\]
03

Simplify the Distance Calculation

First calculate \((2\sqrt{3} + 1)^2 = 4 \cdot 3 + 4\sqrt{3} + 1 = 13 + 4\sqrt{3}\)Then calculate \((2 - \sqrt{3})^2 = 4 - 4\sqrt{3} + 3 = 7 - 4\sqrt{3}\)Now simplify the distance:\[d = \sqrt{(13 + 4\sqrt{3}) + (7 - 4\sqrt{3})} = \sqrt{20}\]Therefore, the distance is:\[d = 2\sqrt{5}\]
04

Conclusion

Thus, the distance between the points \((2, \frac{2 \pi}{3})\) and \((4, \frac{\pi}{6})\) is \(2\sqrt{5}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Coordinates
Polar coordinates provide a different way of defining the position of a point in a plane, using a radius and an angle. Imagine a point that is a certain distance from a central origin. In polar coordinates, this distance is called the radius, and it originates from a reference direction, usually the positive x-axis. The angle, measured in radians, shows how far the point deviates from this reference line in a counterclockwise direction.
  • Typical notation for a polar coordinate is \( (r, \theta) \), where \( r \) is the radius and \( \theta \) is the angle.
  • Polar coordinates are particularly useful in scenarios involving circular and rotational symmetry.
  • A challenge with polar coordinates is that multiple \( \theta \) angles can point to the same location, making unique definitions important.
Cartesian Coordinates
Cartesian coordinates provide another approach to describe a point's position in a plane, using two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point is defined as \( (x, y) \), where \( x \) and \( y \) represent distances along the respective axes.
  • These coordinates are most common for straightforward plotting and many mathematical applications.
  • Cartesian coordinates make it easy to calculate distances and angles between points with straightforward formulas.
  • They serve as a natural foundation for geometry and graphing on a plane.
Distance Formula
The distance formula helps calculate the distance between two points in the Cartesian coordinate plane. It's derived from the Pythagorean theorem and provides a precise way to compute how far apart points are along a straight line.
To find the distance \( d \) between points \( (x_1, y_1) \) and \( (x_2, y_2) \):
  • Substitute the coordinates into the formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
  • Compute the differences \( x_2 - x_1 \) and \( y_2 - y_1 \)
  • Square these differences and sum them up.
  • Take the square root of this sum to get \( d \).
Using the distance formula ensures consistency and accuracy in determining the spatial separation between any two points.
Coordinate Conversion
Coordinate conversion between polar and Cartesian systems is essential when switching descriptions of a point between these two lenses. Depending on the context of a problem, one coordinate system may provide significant advantages over the other.
For converting polar to Cartesian coordinates:
  • Use the equations \( x = r \cdot \cos(\theta) \) and \( y = r \cdot \sin(\theta) \)
  • Here, \( r \) is the radius and \( \theta \) is the angle in radians.
  • Plug in the polar coordinates to get the corresponding \( (x, y) \) values.
For converting Cartesian to polar coordinates:
  • Use \( r = \sqrt{x^2 + y^2} \) to find the radius.
  • For the angle, use \( \theta = \text{atan2}(y, x) \), which correctly handles the signs of \( x \) and \( y \) to determine the quadrant.
Understanding and using coordinate conversion effectively allows one to leverage the strengths of both systems for various mathematical and practical applications.

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Most popular questions from this chapter

Graph the parametric equations using the given range for the parameter t. In each case, begin with the standard viewing rectangle and then make adjustments, as necessary, so that the graph utilizes as much of the viewing screen as possible. For example, in graphing the circle given by \(x=\cos t\) and \(y=\sin t,\)it would be natural to choose a viewing rectangle extending from -1 to 1 in both the \(x\) - and \(y\) -directions. \(x=\sin (0.8 t+\pi), y=\sin t, \quad 0 \leq t \leq 10 \pi\) (Bowditch curve)

Compute each angle of the given triangle. Where necessary, use a calculator and round to one decimal place. $$a=36, b=77, c=85$$

Determine a polar equation for the circle satisfying the given conditions. The radius is \(6,\) and the polar coordinates of the center are: (a) \(\left(-3, \frac{5 \pi}{4}\right)\) (b) \(\left(-2, \frac{\pi}{4}\right)\)

Assume that the vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c},\) and \(\mathbf{d}\) are defined as follows: $$\mathbf{a}=\langle 2,3\rangle \quad \mathbf{b}=\langle 5,4\rangle \quad \mathbf{c}=\langle 6,-1\rangle \quad \mathbf{d}=\langle-2,0\rangle$$ Compute each of the indicated quantities. $$|\mathbf{a}+\mathbf{b}|^{2}+|\mathbf{a}-\mathbf{b}|^{2}-2|\mathbf{a}|^{2}-2|\mathbf{b}|^{2}$$

Graph the parametric equations using the given range for the parameter t. In each case, begin with the standard viewing rectangle and then make adjustments, as necessary, so that the graph utilizes as much of the viewing screen as possible. For example, in graphing the circle given by \(x=\cos t\) and \(y=\sin t,\)it would be natural to choose a viewing rectangle extending from -1 to 1 in both the \(x\) - and \(y\) -directions. \(x=4 \cos t, y=3 \sin t, 0 \leq t \leq \pi / 2\) (one-quarter of an ellipse)
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