/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 Convert the point from cylindric... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 3

Convert the point from cylindrical coordinates to rectangular coordinates. \((2, \pi / 3,2)\)

Short Answer

Expert verified
In rectangular coordinates, the point \((2, \pi / 3, 2)\) is \((1, \sqrt{3}, 2)\).

Step by step solution

01

Identify the cylindrical coordinates

The cylindrical coordinates given are \(r = 2\), \(\theta = \pi / 3\), and \(z = 2\).
02

Calculate the X-coordinate

The formula for converting cylindrical to rectangular coordinates is \(x = r \cos(\theta)\). Substitute the given values into this formula to find the x-coordinate: \(x = 2 \cos(\pi / 3)\).\nBy calculating it we get \(x = 1\)
03

Calculate the Y-coordinate

Similarly, the formula for y-coordinate is \(y = r \sin(\theta)\). Substitute the values: \(y = 2 \sin(\pi / 3)\).\nUpon calculation, we find that \(y = \sqrt{3}\)
04

Identify the Z-coordinate

The z-coordinate in the cylindrical system is the same as the z-coordinate in the rectangular system. Therefore, \(z = 2\).\n

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Coordinate Conversion
Converting between different coordinate systems is an essential skill in mathematics and engineering. It allows us to understand points and objects from various perspectives.
In this context, we focus on moving from cylindrical to rectangular coordinates.

Here's why this is useful:
  • Different systems serve different purposes. Cylindrical coordinates are great for problems involving symmetry around an axis, while rectangular coordinates are widely used for general purposes.
  • Transitioning between these systems can simplify calculations and provide clearer insights.
To convert cylindrical coordinates \(r, \theta, z\) to rectangular coordinates, use these formulas:
- \(x = r \cos(\theta)\)
- \(y = r \sin(\theta)\)
- \(z = z\)
This approach lets us decompose complex problems into manageable pieces, leveraging the strengths of each coordinate system.
Cylindrical Coordinates
Cylindrical coordinates extend polar coordinates by adding a height element.
They are represented as \(r, \theta, z\) where:
  • \(r\) is the radial distance from the origin.
  • \(\theta\) is the angle in the xy-plane from the positive x-axis.
  • \(z\) is the height or depth from the xy-plane.
This system is incredibly useful when dealing with rotations or variations around an axis, like spirals or cylinders.
It simplifies equations involving circular symmetry. For example, consider a point in cylindrical coordinates \(2, \pi/3, 2\):
  • \(r = 2\) tells us how far the point is from the origin in the radial direction.
  • \(\theta = \pi/3\) describes its angle in the plane.
  • \(z = 2\) indicates its height above the plane.
These intuitions help us visualize and understand three-dimensional problems easily.
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, describe points in space using perpendicular axes.
They are expressed as \(x, y, z\):
  • \(x\) is the horizontal distance from the yz-plane.
  • \(y\) is the vertical distance from the xz-plane.
  • \(z\) is the depth from the xy-plane.
This coordinate system is probably the most familiar and is widely used in analytical geometry and physics.
It provides a straightforward way to describe the position of points in a two-dimensional or three-dimensional space.
When converting from cylindrical coordinates like \(2, \pi/3, 2\) to rectangular coordinates, we see:
  • \(x = 1\), calculated as \(2 \cos(\pi/3)\).
  • \(y = \sqrt{3}\), calculated as \(2 \sin(\pi/3)\).
  • The \(z\)-coordinate remains unchanged, \(z = 2\).
Understanding rectangular coordinates allows us to appreciate spatial relationships and easily perform various calculations.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal to \(\mathbf{w}\), then \(\mathbf{u}+\mathbf{v}\) is orthogonal to \(\mathbf{w}\).

Sketch the solid that has the given description in cylindrical coordinates. \(0 \leq \theta \leq \pi / 2,0 \leq r \leq 2,0 \leq z \leq 4\)

Find the distance between the point and the line given by the set of parametric equations.\((4,-1,5) ; \quad x=3, \quad y=1+3 t, \quad z=1+t\)

Sketch the vector \(v\) and write its component form. \(\mathrm{v}\) lies in the \(x z\) -plane, has magnitude 5 , and makes an angle of \(45^{\circ}\) with the positive \(z\) -axis.

Find a set of parametric equations for the line of intersection of the planes.\(6 x-3 y+z=5\) \(-x+y+5 z=5\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.