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Chapter 1: Problem 41

For the following exercises, the rectangular coordinates \((x, y, z)\) of a point are given. Find the spherical coordinates \((\rho, \theta, \varphi)\) of the point. Express the measure of the angles in degrees rounded to the nearest integer.\((4,0,0)\)

Short Answer

Expert verified
The spherical coordinates are \\( (4, 0^\circ, 90^\circ) \\).

Step by step solution

01

Understand the Conversion

To convert from rectangular coordinates \(x,y,z\) to spherical coordinates \(\rho, \theta, \varphi\), we use the following formulas:- \(\rho = \sqrt{x^2 + y^2 + z^2}\)- \(\theta = \arctan\left(\frac{y}{x}\right)\) (in degrees)- \((\varphi = \arccos\left(\frac{z}{\rho}\right)\) (in degrees).These formulas will allow us to translate the given point into spherical coordinates.
02

Calculate 蟻

Insert the given rectangular coordinates into the formula for \(\rho\):\[\rho = \sqrt{4^2 + 0^2 + 0^2} = \sqrt{16} = 4\]Thus, \ \rho = 4 \.
03

Calculate 胃

Since \(y=0\) and \(x=4\), substitute these values into the formula for \(\theta\):\[\theta = \arctan\left(\frac{0}{4}\right) = \arctan(0) = 0^\circ\]Therefore, \ \theta = 0^\circ \.
04

Calculate 蠒

Use the calculated \(\rho\) and given \(z\) to find \(\varphi\):\[\varphi = \arccos\left(\frac{0}{4}\right) = \arccos(0) = 90^\circ\]So, \ \varphi = 90^\circ \.
05

Write Final Spherical Coordinates

Now that all components are calculated, the spherical coordinates are \(\rho = 4\), \(\theta = 0^\circ\), and \(\varphi = 90^\circ\). Thus, the spherical coordinates of the point are \( (4, 0^\circ, 90^\circ) \).

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Key Concepts

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

Rectangular Coordinates
Rectangular coordinates are often referred to as Cartesian coordinates. They allow us to specify the position of a point in a three-dimensional space using three values: \((x, y, z)\).
Each of these values represents a position along one of the three mutually perpendicular axes, often called the x-axis, y-axis, and z-axis.
  • The x-coordinate specifies the position along the x-axis (horizontal line).
  • The y-coordinate indicates the position along the y-axis (vertical line).
  • The z-coordinate provides the position along the z-axis (depth).
This system is modeled like a three-dimensional grid. When combined, these three coordinates can pinpoint any location in a 3D space. Rectangular coordinates are straightforward and very useful in graphing equations, exploring geometric shapes, and solving algebraic problems.
Angle Measurement
When converting between coordinate systems, angle measurement plays an essential role, especially when using spherical coordinates. In the context of spherical coordinates, angles are measured in degrees.
  • \(\theta\) represents the angle in the xy-plane from the positive x-axis direction, akin to direction on a flat plane.
  • \(\varphi\) is the angle between the z-axis and the vector from the origin to the point. This outlines how steep or flat the angle is in three-dimensional space.
These angles allow us to express a position's orientation relative to the origin. Calculating these involves trigonometric functions such as \(\arctan\) and \(\arccos\), converting linear distances into angular separations. Accurate angle measurement is vital for a precise representation of point location in spherical coordinates.
Trigonometry
Trigonometry, the study of angles and their relationships, is indispensable when dealing with spherical coordinates. It provides the tools for converting between different coordinate types, involving calculations like \(\arctan\) and \(\arccos\).
  • \(\arctan\left(\frac{y}{x}\right)\) is used to calculate the angle \(\theta\), signifying the direction of the point in the plane formed by x and y.
  • \(\arccos\left(\frac{z}{\rho}\right)\) gives \(\varphi\), indicating how far the point is above or below the xy-plane.
These functions relate the triangle sides to the angles, converting distances into orientations. Understanding these relationships is crucial for mastering the conversion from rectangular to spherical coordinates, ensuring the point's 3D location is represented appropriately.

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Most popular questions from this chapter

Two forces \(\mathbf{F}_{1}\) and \(\mathbf{F}_{2}\) are represented by vectors with initial points that are at the origin. The first force has a magnitude of \(20 \mathrm{lb}\) and the terminal point of the vector is point \(P(1,1,0)\). The second force has a magnitude of \(40 \mathrm{lb}\) and the terminal point of its vector is point \(Q(0,1,1) .\) Let \(\mathbf{F}\) be the resultant force of forces \(\mathbf{F}_{1}\) and \(\mathbf{F}_{2}\). a. Find the magnitude of \(\mathbf{F}\). (Round the answer to one decimal place.) b. Find the direction angles of \(\mathbf{F}\). (Express the answer in degrees rounded to one decimal place.)

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