Geometric Definition and 790 Threads

Three questions: 1) Anyone have a pedagogic answer to why the image formed by a thin lens is inverted (i.e. upside-down)? I realize that to get a focused image, the lens has to converge the rays one way or another, so then, eventually, the rays have to cross the optic axis, which in turn...

The Fibonacci numbers seem intimately connected with geometry. Prime numbers appear to avoid geometrics, however. Can you give some counterexamples of this latter statement?

Hi, I am trying to understand what Geometric Progression Weighted Average (GPWA) is in the context of the calculation of the US Dollar Index (USDX). I understand what a weighted average is but don’t understand what a GPWA is and when one should use it. The following equation shows how the...

1. Sum the Geometric Series 1-x+x2-x3+x4 and hence simplify the function [f(x)]4 = 1 - x5 1-x+x2-x3+x4 Homework Equations 3. Not sure I quite get understand this properly, as my attempt doesn't seem quite right. Basically I've gotten...

Homework Statement A Mooney rhomb is a quadrilateral prism that converts linear light to circular polarized light. The question is here...

Homework Statement (2+x)+(2+x)^2+(2+x)^3 + ... Homework Equations The Attempt at a Solution Ive found that the l r l < 1 the r of this equation is (2 + x) so we have -1 < 2 + x < 1 The values of x where the series coverges is -3 < x < -1 Is this correct...

If given the values -1, 5, 2 in this sequence, what would be the missing term to make this a geometric series? Also, what would the sum of this geometric series be?

Homework Statement 1+(x+1)+(x+1)^2+(x+1)^3 + ... if lx+1l < 1 Homework Equations Sn=a/1-r The Attempt at a Solution My attempt: so I have a = x+1 and r = x+1 from there i get x+1/1-(x+1) which is x+1/1-x-1 from there x+1/-x multiply by the reciprocal my...

There is a still-water lake and air interface. Light travels from the water to the air so that the incident angle is also a critical angle, making it so that the light runs along the surface of the water. Considering that this ray is reversible (air to water), a fish looking up at the surface...

Homework Statement Hi! I just used MATLAB to find the eigenvalues and eigenvectors of A=[0 -1; 1 0] I obtained the eigenvalues of 0 +/- i and eigenvectors of v(1) = [ 0.7071; 0 - 0.7071i] and v(2) = [ 0.7071; 0 + 0.7071i] Homework Equations I'm having trouble interpreting these results in...

Homework Statement The first 3 of a geometric progression are as follows: 2^{n}, 2^{n+1}, 2^{n+2} Find n Homework Equations For this kind of question, the only possibly helpful equation I can think of would be: T_{1}=a The Attempt at a Solution The problem I have is that at first...

Homework Statement Suppose A and B are similar matrices, and that (mu) is an eigenvalue of A. We know that (mu) is also an eigenvalue of B, with the same algebraic multiplicity(proved in class) Suppose that g is the geometric multiplicity of (mu), as an eigenvalue of B. Show that (mu) has...

Ax = U \Sigma V^T x (A is an m by n matrix) I understand the first two steps, 1) V^T takes x and expresses it in a new basis in R^n (since x is already in R^n, this is simply a rotation) 2) \Sigma takes the result of (1) and stretches it The third step is where I'm a bit...

Vectors A and B each lie in the x-y plane. The magnitude of A + B equals the magnitude of A - B. Find the geometric relation between vectors A and B. (Hint: Express the vectors in unit-vector notation. Use your knowledge of dot products.) I really don't even know what this problem is asking...

The question from my textbook was "Give a complete proof that the largest triangle that can be inscribed in a circle is an equilateral triangle ", largest meaning that with the greatest area. I found a proof that finds the area of the general triangle in terms of the angles, and through...

I'm having difficulty understanding the geometry of a spacetime split as it applies to geometric algebra. I understand vectors, bivectors, and trivectors and I understand geometric products and wedge products. My impression is that a spacetime split let's you decompose an electromagnetic field...

In words, the sum of a geometric sequence can be written out to say "the first term divided by (1 minus the common ratio)". Does the first term also apply when the series starts with some other number n other than 1 (like 2 or 3, etc)? In other words, the first term is when n = some other number...

I've been stuck with this problem: An ellipse can be defined as 1) locus of points for which is constant the sum of the distances from two fixed points (foci) 2) locus of points for which is constant the ratio between the distances from a fixed point (focus) and a fixed line (directrix)...

Homework Statement Does the series from n=1 to infinity of (2)/(n^2-1) converge or diverge? If it converges, find the sum. Homework Equations The Attempt at a Solution I can see right away that the series converges by a limit comparison test by looking at the series. However, to find the sum...

Hi, I'm having trouble finding the sequence's total sum from a formula concerning Geometric Series. I've been using a calculator to find and manually input all of the terms into a table in Microsoft Excel and adding them all up at the end. The formula that I was given was...

Hi guys, In a geometric progression, the second term exceeds the first term by 20 and the fourth term exceeds the second term by 15. Find the possible values of the first term. ar - a = 20 ar3 - ar = 15 ar = 20 + a (20 + a)3 - (20 + a) = 15 But this method seem to be wrong. Any help?

Homework Statement On an exam question, although I can not remember the details, it gave us a table that looks similar to the table below. http://img403.imageshack.us/img403/9703/geoseqpr4.png [/URL] The question tells us that the graph is a geometric sequence, and says to use a formula to...

http://img117.imageshack.us/img117/5258/w1vg1.th.jpg http://img84.imageshack.us/img84/3151/w2px1.th.jpg See above files (one is the question and one is the answer) I can to the whole question, other than the last part - for part (f), why are we concerned with the sum to infinity of...

What the heck? The minimum monthly payment for a credit card is the larger of $5 or 1/25 of the outstanding balance. If the balance is less than $5, then the entire balance is due. If you make only the minimum payment each month, how long will it take to pay off a balance of $200? Clearly...

Homework Statement A mortgage is taken out for £80,000. It is to be paid by annual instalments of 5000with the first payment being made at the end of the first year that the mortgage was taken out. Interest of 4% is then charged on any outstanding debt. Find the total time taken to pay off the...

[SOLVED] Geometric Algebra: Signs of electromagnetic field tensor components? Here's a question that may look like an E&M question, but is really just a geometric algebra question. In particular, I've got a sign off by 1 somewhere I think and I wonder if somebody can spot it. PF isn't...

Hey, (sin A + sin B + sin C)/3 >= \sqrt[3]{}(sin A*sin B*sin C) I know this is true by Arithmetic mean always greater than geometric mean... but is there any other way of proving this?

I'd like the geometric meaning of the Mean Value Theorem. Say for instance I had a function of velocity that varied as t^{3} + 3t^{2} + 3t +1. I consider the interval [0,4]. So by MVT, I have a number c in [0,4] such that f'(c)(4) = f(4) - f(0). What does that mean? That there is an...

[SOLVED] geometric algebra: longitude and latitude rotor ordering? Was playing around with what is probably traditionally a spherical trig type problem using geometric algebra (locate satellite position using angle measurements from two well separated points). Origin of the problem was just me...

Homework Statement If a plane mirror is hung at an angle of 30 degrees from the vertical toward the viewer. what is the minimum length that will allow the viewer to see the full body image if the line of vision can not be depressed below the horizontal. you may assume that the eyes are at the...

Homework Statement I'm not interested in the proof of this statement, just its geometric meaning (if it has one): Suppose T \in L(V) is self-adjoint, \lambda \in F, and \epsilon > 0. If there exists v \in V such that ||v|| = 1 and || Tv - \lambda v || < \epsilon, then T has an...

Hi, I have a problem with computing this geometric series. Homework Statement I have to compute \sum_{i=0}^\infty{(\frac{1}{2z})^{2k}} + \sum_{i=0}^\infty{(\frac{1}{3z})^{2k+1}}. It's for computing the z-transform of f[k]=0 for k<0 f[k]=(\frac{1}{2})^k for k=0,2,4,6...

Homework Statement Homework Equations The Attempt at a Solution I don't get what I am doing wrong here, I have attached my solution below. The solution manual have their answer as 3e/(3-e). Thanks!

Homework Statement a) Two friends, Jon and Bob, are sharing a loaf of bread. Jon eats half of the loaf, then Bob eats half of what remains, then Jon eats half of what remains and so on. How much of the loaf did each of them eat? b)Jon is hungrier and eats 2/3 of the loaf, then Bob eats half...

helo this is a homework problem i got in math 30 pure i got an answer but i would like to know how to get it by using a formula? The exercise gose like this: Initially, a pendulum swings through an arc of 2feet. On each successive swing,the length of the arc is 0.9 of the previous length...

I getting far enough into my geometric algebra books now that I'm at linear transformations, including the result showing how a linear transformation inverse can be expressed directly as a geometric product, using the adjoint and pseudoscalar multiplication, instead of using matrix inversion...

Hi In trying to calculate the following sum: \sum_{i=1}^n{i^2} I found the following expansions: \sum_{i=1}^p \sum_{j=0}^{i-1} (-1)^j(i-j)^p {n+p-i+1\choose n-i} {p+1\choose j} My question is: is there an easier or more intuitive way to compute the limit of the sum above?

My differential forms book (Flanders/Dover) defines an inner product on wedge products for vectors that have a defined inner product, and uses that to define the hodge dual. That wedge inner product definition was a determinant of inner products. I don't actually have that book on me right...

I was given an ap problem in class, specifically: Problem 2:http://www.collegeboard.com/prod_downloads/ap/students/calculus/b_calculus_bc_frq_03.pdf I was able to do it just fine, but then I had the idea to try and solve it geometrically/algebraically. However, I haven't been able to come...

Given the parametric representation of two planes, through points P and Q respectively x = P + \alpha u + \beta v y = Q + a w + b z Or, alternately, with u \wedge v = A, and w \wedge z = B x \wedge A = P \wedge A y \wedge B = Q \wedge B It's easy enough to find...

Hi all, We are now going to do a kind of experiment in which we use a plumb bob to identify the vertical direction. But I think, the pumb string will point to the center of mass, while we need the geometric center . I mean the COM of the Earth is diffent from the geometric center and it...

geometric algebra cross product -------------------------------------------------------------------------------- Homework Statement [/b] my text (Geometric Algebra for Physicists, by Doran and Lasenby), p. 69, deals with rotating frame {fsubk} (I assume in 3D) d/dt (fsubk) = omega X...

Homework Statement [/b] my text (Geometric Algebra for Physicists, by Doran and Lasenby), p. 69, deals with rotating frame {fsubk} (I assume in 3D) d/dt (fsubk) = omega X fsubk omega being angular velocity then omega X fsubk = (-I omega) dot fsubk = fsubk dot (I omega), where...

Does anyone know how to evaluate S_n = \sum_{i=0}^{n-1} i2^i I tried the following. Let r = 2, and figure out the terms in S_n - rS_n Unlike with a regular geometric series, this does not make all but two of the terms disappear. But it does make all but one of the terms turn into a...

Homework Statement A river is 2 km wide and flows at 6 km/h. A motor boat that has a speed of 20 km/h in still water heads out from one bank perpendicular to the current. A marina lies directly across the river on the opposite bank. Use Geometric Vectors to solve this problem. a. How far...

[SOLVED] Geometric Vectors Homework Statement Find the magnitude and the direction (to the nearest degree) of the resultant of each of the following systems of forces. a. forces of 3 N and 8 N acting at an angle of 60º to each other - use geometric vectors to solve this problem...

Let A and B be similar matrices. Prove that the geometric multiplicities of the eigenvalues of A and B are the same. Some help I have gotten so far but still don't know how to proceed from there: To prove that the geometric multiplicities of the eigenvalues of A and B are the same, we can...

Homework Statement Determine the sum of the series: \sum^{infinity}_{K=10} \frac{7}{e^(3k+2)} Homework Equations The Attempt at a Solutionlimit n->infinity of sn=\sum^{n}_{K=10} \frac{7}{e^(3k+2)}=\frac{7}{e^(32)}+\frac{7}{e^(35)}...\frac{7}{e^(3n+2)} This series does not exactly fit a...

Homework Statement See images. . . Can you please correct my answers if I get some of them wrong? Homework Equations There aren't really any relevant equations to solve the problemsThe Attempt at a Solution See images. . .

I am trying to teach myself and often get stuck. Right now I've come across a . B = 1/2(aB-Ba) where a is a vector and B is a bivector. what's confusing me is that it seems to require a change in the definition of geometric product as the sum of a symmetrical inner product and...