A Cartesian coordinate system (UK: , US: ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis (plural axes) of the system, and the point where they meet is its origin, at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.
One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, n Cartesian coordinates (an element of real n-space) specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes.
The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4.
Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing.
Hello,
This question, which I found in various electricitiy and magnetism books (e.g. Introduction to electrodynamics grif.).
There are many variations of this question, I am mainly interested in the following setup of it:
-Suppose there is a charged disk of radius R lying in the xy-plane, and...
How to prove that every quadric surface can be translated and/or rotated so that its equation matches one of the six types of quadric surfaces namely 1) Ellipsoid
2)Hyperboloid of one sheet
3) Hyperboloid of two sheet 4)Elliptic Paraboloid
5) Elliptic Cone 6) Hyperbolic Paraboloid
The...
Homework Statement
[/B]
An object of m-mass is to be thrown from xy-plane with an initial velocity ##\mathbf v_0 = v_0\mathbf e_z \, (v_0 > 0)## to a force field ##\mathbf F = -F_0 e^{-z/h}\mathbf e_z\,## , where ##F_0, h > 0## are constants. By what condition does the object return to...
Homework Statement
Calculate the line integral ° v ⋅ dr along the curve y = x3 in the xy-plane when -1 ≤ x ≤ 2 and v = xy i + x2 j.
Note: Sorry the integral sign doesn't seem to work it just makes a weird dot, looks like a degree sign, ∫.2. The attempt at a solution
I have to write something...
Homework Statement
Hello!
Last week I have came here for the help related to this problem. I am creating a new thread to describe the issue more precisely. I will be grateful for your help and explanation.
I post the explanation for the book first accompanied by attached pictures, and below I...
Homework Statement
Essentially it gives the potential above the xy-plane as and I am tasked with verifying it satisfies laplace's equation, determining the electric field, and describing the charge distribution on the plane.
Homework Equations
then
The Attempt at a Solution
As far as I...
Hello, I picked up a challenging problem (at least to me) and I'm having difficulties.
1. Homework Statement
An object moves in xy-plane from point O = (0; 0) to point A = (1 m; 0) and from there to point B = (1 m; 2 m). All this time when the object moves a force \vec F = ax2\vec i + by\vec...
Homework Statement [/b]
"Find an equation for the line through the point P = (1, 0, −3) and perpendicular
to the xy-plane,"
obviously this includes vector <0, 0, 1>
I am in Calc III and need help understanding how to do this TYPE of problem. Please include step-by-step instructions and...
Part of a chapter review problem.
Say you have a sphere centered at the origin and of radius 'a'.
And you have a (ice-cream) cone which has it's point at the origin and phi equal to ∏/3.
How do I find the equation of their intersection? Which is the projection onto the xy plane...
Homework Statement
Consider a circular ring of wire of radius a that resides in the x-y plane through the origin. The center of the ring coincides with the origin and you can regard the thickness of the wire to be infinitesimal.
a. Given that a current I flows in the ring, find an...
Find the surface area of that portion of the sphere x^2 + y^2 + z^2 = a^2 that is above xy-plane and within the cylinder x^2 + y^2 = b^2 , 0 < b < a
Solution.. i try to find fx and fy..
http://imageshack.us/f/594/33049204.jpg/"
how am i going to proceed?
Homework Statement
So, the problem is this:
Find all values of t such that r'(t) is parallel to the xy-plane.
And my equation is:
r(t)=(Squareroot(t+1) , cos(t), t4-8t2)
Homework Equations
Well, I will definitely have to know how to take the dirivative of the given vector valued...
Homework Statement
[PLAIN]http://img293.imageshack.us/img293/9080/omgay.jpg
Homework Equations
?
The Attempt at a Solution
?? I'm already lost at where to begin.
What does it mean if a line in R^3 is parallel to the xy-plane but not to any of the axes. I really don't know what this means in terms of how the parametric and symmetric equations of the line should look. Please help. Thanks.
Homework Statement
So my question is: what is the volume of the region R between the paraboloid 4-x^2-y^2 and the xy-plane?
Homework Equations
I know how to solve it, it is a triple integral, but how do you find the limits of integration?
The Attempt at a Solution
Do I set x=0...
[SOLVED] diff. paths of a Force in xy-plane
Homework Statement
A force acting on a particle in the xy-plane is given by
\vec{F} = (2yi + x^2j) where x and y are in meters.
The particle moves from the origin to a final position having coordinates
x = 5.00m and y = 5.00m...
Sketch the plane curve in the xy-plane and find its length over the given interval:
r(t) = (6t-3)i + (8t+1)j on [0,3]
Here's what I've got so far:
r'(t) = 6i + 8j
llr'(t)ll = sqrt of 6^2+8^2 = 10
s = integral 0-3 10dt
= 10x ]0 to 3
= [30-0]
= 30I just need help on how to sketch this plane...
Question:
A particle of charge 4.96 nC is placed at the origin of an xy-coordinate system, and a second particle of charge -1.95 nC is placed on the positive x-axis at x = 3.99 cm. A third particle, of charge 6.04 nC is now placed at the point x = 3.99 cm, y = 3.05 cm.
Part A
Find the...
A particle of mass M is free to move in the horizontal plane(xy-planne here). It is subjected to force \vec F = -k\left(x\hat i + y\hat j\right), where 'k' is a positive constant.
There are two questions that have been asked here:
1] Find the potential energy of the particle.
\vec \nabla...
Consider the solid that lies above the square (in the xy-plane) R= [0,1] X [01]
and below the elliptic paraboloid z= 64 -x^2 +4xy -4y^2
Estimate the volume by dividing R into 9 equal squares and choosing the sample points to lie in the midpoints of each square.
i'm not sure how you...