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Chapter 11: Problem 3

Change the following from cylindrical to Cartesian (rectangular) coordinates. (a) \((6, \pi / 6,-2)\) (b) \((4,4 \pi / 3,-8)\)

Short Answer

Expert verified
(a) \((3\sqrt{3}, 3, -2)\); (b) \((-2, -2\sqrt{3}, -8)\).

Step by step solution

01

Understanding the Relationship between Coordinates

In cylindrical coordinates, a point in space is represented as \((r, \theta, z)\), where \(r\) is the radial distance from the \(z\)-axis, \(\theta\) is the angular position measured from the positive \(x\)-axis, and \(z\) is the height above the \(xy\)-plane.In Cartesian coordinates, a point is given as \((x, y, z)\). The conversion formulas from cylindrical to Cartesian coordinates are:\[x = r \cos \theta, \quad y = r \sin \theta, \quad z = z.\]
02

Convert Part (a) from Cylindrical to Cartesian Coordinates

Given cylindrical coordinates \((6, \pi / 6, -2)\):- Calculate \(x\): \[x = 6 \cos \left(\frac{\pi}{6}\right) = 6 \times \frac{\sqrt{3}}{2} = 3 \sqrt{3}. \]- Calculate \(y\): \[y = 6 \sin \left(\frac{\pi}{6}\right) = 6 \times \frac{1}{2} = 3. \]- The \(z\) coordinate remains the same as \(z = -2\).Thus, the Cartesian coordinates are \((3 \sqrt{3}, 3, -2)\).
03

Convert Part (b) from Cylindrical to Cartesian Coordinates

Given cylindrical coordinates \((4, 4\pi / 3, -8)\):- Calculate \(x\): \[x = 4 \cos \left(\frac{4\pi}{3}\right) = 4 \times \left(-\frac{1}{2}\right) = -2. \]- Calculate \(y\): \[y = 4 \sin \left(\frac{4\pi}{3}\right) = 4 \times \left(-\frac{\sqrt{3}}{2}\right) = -2\sqrt{3}. \]- The \(z\) coordinate remains the same as \(z = -8\).Thus, the Cartesian coordinates are \((-2, -2\sqrt{3}, -8)\).

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Key Concepts

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

Cylindrical Coordinates
Cylindrical coordinates are a way to describe the position of a point in a three-dimensional space using three parameters: \(r\), \(\theta\), and \(z\). This system is particularly useful when dealing with problems that have a natural circular symmetry, such as those encountered in physics and engineering.
  • \(r\) represents the radial distance from the \(z\)-axis to the point. It's similar to the radius of a circle.
  • \(\theta\) is the angular position. It is measured in radians and represents the angle from the positive \(x\)-axis on the \(xy\)-plane.
  • \(z\) is the height of the point above the \(xy\)-plane, similar to the \(z\) in Cartesian coordinates.
This coordinate system is very helpful when dealing with problems involving rotations or cylindrical shapes, as it aligns well with their geometry.
Cartesian Coordinates
In contrast to cylindrical coordinates, Cartesian or rectangular coordinates describe a point using a straightforward grid system. Here, each point is given as \(x\), \(y\), \(z\), akin to a classic map or graph. This system is simple and intuitive, making it a popular choice for many mathematical applications.
  • \(x\) is the horizontal position along the \(x\)-axis.
  • \(y\) is the vertical position along the \(y\)-axis.
  • \(z\) denotes the position along the \(z\)-axis, indicating height or depth.
This method is perfect for problems that require precise locations or involve non-circular shapes, offering a clear visualization of spatial relationships.
Conversion Formulas
To convert between cylindrical and Cartesian coordinates, one must use a set of conversion formulas. These formulas translate the radial and angular components into the familiar rectangular coordinate system.
  • The formula for \(x\) is \(*x = r \cos \theta*\). This relates the radial distance and angle to the horizontal position.
  • For \(y\), use \(*y = r \sin \theta*\), which converts radial distance and angle to vertical position.
  • The \(z\) coordinate remains unchanged, as both cylindrical and Cartesian systems share the same height dimension.
These conversion formulas allow for easy switching between the two systems, catering to different problem-solving scenarios and enhancing geometric understanding.
Trigonometric Functions
Trigonometric functions play a crucial role in the conversion process between cylindrical and Cartesian coordinates. These functions help in computing the necessary translations of angles and distances.
  • **Cosine (\(\cos\))** is used when translating angles into an \(x\) component. It's part of the conversion formula for \(x\) as \(x = r \cos \theta\).
  • **Sine (\(\sin\))** relates angles to a \(y\) component, appearing in the formula \(y = r \sin \theta\).
  • These functions operate based on a right-angled triangle, making them extremely efficient at calculating dimensions for circular movements.
Understanding these functions is key to mastering coordinate conversions, as they bridge the gap between circular (cylindrical) and grid (Cartesian) systems.

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Most popular questions from this chapter

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