/*! 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 24 In a Cartesian system, \(A\) and... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 24

In a Cartesian system, \(A\) and \(B\) are the points \((0,0,-1)\) and \((0,0,1)\) respectively. In a new coordinate system a general point \(P\) is given by \(\left(u_{1}, u_{2}, u_{3}\right)\) with \(u_{1}=\frac{1}{2}\left(r_{1}+r_{2}\right), u_{2}=\frac{1}{2}\left(r_{1}-r_{2}\right), u_{3}=\phi ;\) here \(r_{1}\) and \(r_{2}\) are the distances \(A P\) and \(B P\) and \(\phi\) is the angle between the plane \(A B P\) and \(y=0\)

Short Answer

Expert verified
Transforming with given functions validates the distances from A and B to P under new coordinates.

Step by step solution

01

- Identify the given points

Identify the coordinates of points A and B. Given: A(0,0,-1) and B(0,0,1) in the Cartesian system.
02

- Define the distance functions

Define r_1 and r_2, where r_1 is the distance from point A to a general point P and r_2 is the distance from point B to point P.
03

- Write the distance formulas

Use the distance formula in 3D: \(r_1 = \sqrt{(u_1 - 0)^2 + (u_2 - 0)^2 + (u_3 + 1)^2}\) and \(r_2 = \sqrt{(u_1 - 0)^2 + (u_2 - 0)^2 + (u_3 - 1)^2}\).
04

- Substitute into the given coordinates

Substitute the given coordinate functions into the distance formulas: \(u_{1} = \frac{1}{2}(r_{1} + r_{2})\), \(u_{2} = \frac{1}{2}(r_{1} - r_{2})\), and \(u_{3} = \phi \).
05

- Simplify the equations

Simplify the expressions to express r_1 and r_2 in terms of u_1 and u_2. This will validate whether the distances are consistent as per the provided transformation.

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.

Cartesian System
The Cartesian system is a coordinate system used to specify points in space using coordinates on perpendicular axes. Typically, it uses three coordinates for 3-dimensional space: \(x\), \(y\), and \(z\). These coordinates represent a point's position relative to three perpendicular lines known as the x-axis, y-axis, and z-axis.
Each coordinate specifies how far the point is from each axis. In mathematical terms, a point \(P\) in a 3D Cartesian system is denoted as \(P(x, y, z)\).
For example, point \(A(0,0,-1)\) means it is on the z-axis, \(-1\) unit below the origin. Similarly, point \(B(0,0,1)\) is \(+1\) unit above the origin on the z-axis.
Distance Formula
The distance formula helps calculate the distance between two points in space. For 3D coordinates, the formula to find the distance \(d\) between points \(P(x_1, y_1, z_1)\) and \(Q(x_2, y_2, z_2)\) is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
This formula helps transform geometric concepts into numerical values. For instance, calculating distances \((r_1\) and \(r_2)\) involves using this formula with the coordinates:
\[ r_1 = \sqrt{(u_1 - 0)^2 + (u_2 - 0)^2 + (u_3 + 1)^2} \]
\[ r_2 = \sqrt{(u_1 - 0)^2 + (u_2 - 0)^2 + (u_3 - 1)^2} \]
Here, \(u_1\), \(u_2\), and \(u_3\) are the coordinates of a general point \(P(u_1, u_2, u_3)\).
3D Coordinates
3D coordinates extend the concept of 2D coordinates into an additional dimension, offering a complete description of points in space. This is useful for analyzing positions and shapes in three-dimensional contexts.
Each point in 3D space is defined as \(P(x, y, z)\), where:
  • \(x\)-coordinate shows the distance from the yz-plane
  • \(y\)-coordinate shows the distance from the xz-plane
  • \(z\)-coordinate shows the distance from the xy-plane

In our problem, a point \(P(u_1, u_2, u_3)\) is defined using new coordinates involving distances and an angle:
  • \(u_{1} = \frac{1}{2}(r_1 + r_2)\)
  • \(u_{2} = \frac{1}{2}(r_1 - r_2)\)
  • \(u_{3}\) is an angle \(\phi \) with respect to a plane

This representation simplifies the transformation between different coordinate systems.
Angle Between Planes
The angle between planes is a crucial concept in 3D geometry. When analyzing points and lines within different planes, it's often necessary to determine the angle at which the planes intersect.
In this case, we need to find the angle \(\phi\) between the plane formed by \(ABP\) and another given plane, which is \({y=0}\):
  • The plane \(ABP\) includes points A, B, and a general point \(P(u_1, u_2, u_3)\)
  • The plane \({y=0}\) is the xz-plane

To evaluate \(\phi\), we use trigonometric relationships and spatial orientation. Knowing \(\phi\) allows better understanding of the spatial relationship between the transformed coordinates and original Cartesian system.

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

Evaluate the Laplacian of a vector field using two different coordinate systems as follows. (a) For cylindrical polar coordinates \(\rho, \phi, z\), evaluate the derivatives of the three unit vectors with respect to each of the coordinates, showing that only \(\partial \hat{\mathbf{e}}_{\rho} / \partial \phi\) and \(\partial \hat{\mathbf{e}}_{\phi} / \partial \phi\) are non-zero. (i) Hence evaluate \(\nabla^{2} \mathbf{a}\) when \(\mathbf{a}\) is the vector \(\hat{\mathbf{e}}_{\rho}\), i.e. a vector of unit magnitude everywhere directed radially outwards and expressed by \(a_{\rho}=1, a_{\phi}=\) \(a_{z}=0\) (ii) Note that it is trivially obvious that \(\nabla \times \mathbf{a}=\mathbf{0}\) and hence that equation \((10.41)\) requires that \(\nabla(\nabla \cdot \mathbf{a})=\nabla^{2} \mathbf{a}\) (iii) Evaluate \(\nabla(\nabla \cdot \mathbf{a})\) and show that the latter equation holds, but that $$ [\nabla(\nabla \cdot \mathbf{a})]_{\rho} \neq \nabla^{2} a_{\rho} $$ (b) Rework the same problem in Cartesian coordinates (where, as it happens, the algebra is more complicated).

Verify by direct calculation that $$ \nabla \cdot(\mathbf{a} \times \mathbf{b})=\mathbf{b} \cdot(\nabla \times \mathbf{a})-\mathbf{a} \cdot(\nabla \times \mathbf{b}) $$

Paraboloidal coordinates \(u, v, \phi\) are defined in terms of Cartesian coordinates by $$ x=u v \cos \phi, \quad y=u v \sin \phi, \quad z=\frac{1}{2}\left(u^{2}-v^{2}\right) $$ Identify the coordinate surfaces in the \(u, v, \phi\) system. Verify that each coordinate surface \((u=\) constant, say) intersects every coordinate surface on which one of the other two coordinates \((v\), say \()\) is constant. Show further that the system of coordinates is an orthogonal one and determine its scale factors. Prove that the \(u\)-component of \(\nabla \times \mathbf{a}\) is given by $$ \frac{1}{\left(u^{2}+v^{2}\right)^{1 / 2}}\left(\frac{a_{\phi}}{v}+\frac{\partial a_{\phi}}{\partial v}\right)-\frac{1}{u v} \frac{\partial a_{v}}{\partial \phi} $$

Use vector methods to find the maximum angle to the horizontal at which a stone may be thrown so as to ensure that it is always moving away from the thrower.

Evaluate the Laplacian of the function $$ \psi(x, y, z)=\frac{z x^{2}}{x^{2}+y^{2}+z^{2}} $$ (a) directly in Cartesian coordinates, and (b) after changing to a spherical polar coordinate system. Verify that, as they must, the two methods give the same result.
See all solutions

Recommended explanations on Physics Textbooks

Medical Physics

Read Explanation

Measurements

Read Explanation

Magnetism

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Fields in Physics

Read Explanation

Physical Quantities And Units

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.