Explain the difference in the uniqueness of the representations of Cartesian coordinates vs polar coordinates?
438064
your question makes my head hurt.
323643
kate1976 has solved the problem using the traditional way, just using the concept of coordinate Geometry.  Usin the cross product of two vectors, the method is different.\n\nDot and Cross Products:\nSuppose [x1,y1] = [ r1 cos(q1), r1 sin(q1)] and \n[x2,y2] = [ r2 cos(q2), r2 sin(q2)] are points in a plane. Then their dot product\n\n[x1,y1].[x2,y2] = x1x2+y1y2  (dot product definition)  ..(1)\n\nand their cross product\n\n[x1,y1].[x2,y2] = x1y2 - y1x2 (cross product definition).(2)\n\nmay be expressed in terms of  trigonometric functions and the angles between the two points, or more precisely their position vectors.\nTo obtain the geometric interpretation,  observe the polar and rectangular ways to multiply the first point by the complex complex conjugate of the second, when both are viewed as complex numbers: That is,\n\n [x1,y1][x2,-y2] = (r1,q1)(r2,-q2)\n\nFrom the equality of two different ways to multiply points in the plane,  observe\n\n[x1x2+y1y2, x1y2 - y1x2]  = (r1r2,q1-q2)\n\nbut\n\n(r1r2,q1- q2) = [ r1r2  cos( q1- q2), r1r2  sin( q1- q2)]\n\nTherefore comparison (equality) of real and imaginary parts yields:\n\nx1x2 + y1y2 = r1r2  cos(q)\n\nand\n\n x1y2 - y1x2 = r1r2  sin( q)\n\n where\n\n q = q1- q2\n\nis the angle between the two points  [x1,y1] = (r1 , q1) and [x2,y2] = ( r2 , q2)\n\n!!!!!!!!!!! Please read q as theta !!!!!!!!!!!\n\nNow proceed using the equations (1) and (2)
908924
Because of (one of) the definition(s) of sin and cos.\nWhich is: take a unit circle, and draw in any radius at some angle theta. Then sin theta = the y value, and cos theta = the x value. Those are obviously less than 1.\n\nOr you could just take a typical right angled triangle - the length of the opposite/adjacent sides are always less than the length of the hypotenuse.
362083
Please use the search function. This question was asked 5 days ago:\n\nA proper polar coordinate system (one that assigns each point in the plane a unique coordinate) will always leave out a ray of points. For example, the standard polar system that identifies the open set (0, oo) X (0, 2Pi) with R^2 - {0} by the function f(r, t) = (rcos(t), rsin(t)) ( = (x, y) ) does not correspond to any points on the positive x-axis and necessarily leaves out the origin (there is no way to include the origin uniquely). This is usually glossed over in a basic calculus or physics course as one actually uses an atlas of polar coordinate charts (this atlas never includes the origin, but at least covers the plane minus the origin); you use whichever chart is convenient for the problem at hand. Some definitions will disregard the need for a homeomorphism and include the positive x-axis by considering a domain that is not open, (0, oo)X[0, 2Pi)
150881
There are several reasons. One of the most basic ones is that this representation provides a "polar" representation of complex numbers. Any complex number z=a+ib can be written as\n\nz= r exp(i theta)\nr= sqrt(a^2+b^2)\ntheta = atan(b/a)\n\nThis helps because, for example, a product of z1 and z2 would give\n\nz1 z2 = R exp(i phi)= r1*r2 exp (theta1 + theta2)\nR=r1*r2\nPhi = theta1 + theta2\n\nSo multiplying by z2 "stretches" z1 by amount r2 and "rotates" z1 by theta2. Such interpretations are convenient in many instances. Also, exponential functions are easier to deal with than regular a+ib type expressions in other cases.
740929
Without a little more context I can't say for sure but, most likely, it's just another way of asking for the vector sum. That's the one vector that is equivalent to adding up the directions and magnitudes of all other vectors in the problem.
819996
T1=m1*a, no matter are there m2 & m3 or not.\nT2=(m1+m2)*a, no matter whether m1 & m2 are connected with a string or are in one lump, the presence of m3 being unimportant too.
738154
In mathematics, the Cartesian coordinate system is used to uniquely determine each point in the plane through two numbers, usually called the x-coordinate and the y-coordinate of the point. To define the coordinates, two perpendicular directed lines (the x-axis or abscissa and the y-axis or ordinate), are specified, as well as the unit length, which is marked off on the two axes (see Figure 1). Cartesian coordinate systems are also used in space (where three coordinates are used) and in higher dimensions.\n\nUsing the Cartesian coordinate system geometric shapes (such as curves) can be described by algebraic equations, namely equations satisfied by the coordinates of the points lying on the shape. For example, the circle of radius 2 may be described by the equation x² + y² = 4
267076
"Cartesian coordinates come from his last name, Descarte."\n\nActually, you are only half right. The Cartesian plane comes from Descarte's Latin name, Cartesius, which is what he used when publishing his proofs. \n\nOnto the question at hand, I'm not even sure what you are asking. What did Rene Descartes use the first letters of? Grammatical abnormalities notwithstanding, this question is very vague. What exactly do you mean?
383124
I believe the book is incorrect as well.\n\nThere are some real-world problems that start and end with numbers that do not belong to the set of Reals.  Consider the Quaternion.  A Quaternion is like a "super imaginary number", it is of the form, a + b*i + c*j + d*k, where i^2 = j^2 = k^2 = -1\n\nThis type of number can be used to represent the position and orientation of an object in 3D.  Since objects do have positions and orientations, it is not unreasonable to represent this with a Quaternion (even though many people don't know what it is).
986763
To convert from cartesian to polar coordinates, the formulas are:\n\nr = √(x² + y²)\nθ = arctan(y/x)\n\nso r = √((-2)² + (3)²) = √(4 + 9) = √13\n\nand θ = arctan(y/x) = arctan(3/-2) = -56.3°\n\nBut you can also add or subtract 360° as many times to theta as you want and you still get the same angle.\n\nSo two possible representations would be:\n\n(√13, -56.3°) and\n(√13, 303.7°)
829392
just a piont of view, but normally to divide something symmetrically you have to do it at the widest piont along the centre line. where as assymetricaly is off the centre line at a narrower piont and therefore not a symmetry,this only applies to poloygons and symmetric shapes.  hope this helps
598810
a regular polygon has the same angles and sides while a irregular one has different sides and angles.
939642
Counterexample.  Let u(x), v(x) be members of {sin x, cos x}.  Both of those are solutions, but you do not get zero by adding one to the other, or by doubling one of them.
749264
Usually there won't be an answer. \n\nFor planes you have:\nz=ax+by+c\nz=dx+ey+f --->\nax+by+c=dx+ey+f\n(a-d)x+(b-e)y=c-f ---> gives a line for an answer.\n\nHowever if you look at two lines, you will have that the x and y coordinates are dependent on each other:\nI.e.\nz=f(x),y=g(x)\nz=h(x),y=j(x)\nSo --- f(x)=h(x) and g(x)=j(x) ---- which is over determined.
998200
Polar\nits use to converse from dimensional to polar.\nexp\n(3,4) --> (5;53)\nits mean 5 for radius and 53degree
669147
number of sides, congurency?, size of angles, distinguishing characteristics