Monday

January 16, 2017
Number of results: 16,433

**Calculus - Orthogonal Trajectories**

Find the orthogonal trajectories of the family of curves: y = k*(e^-x) --------------- so k = y/(e^-x) differentiating we get: 1 = -k(e^-x)*(dx/dy) 1/(dx/dy) = -k(e^-x) dy/dx = -k(e^-x)...substituting for k: dy/dx = -(y/(e^-x))*(e^-x) dy/dx = -y Integral(1/=y)dy = Integral dx...

*June 26, 2007 by COFFEE*

**calc 2**

Find the orthogonal trajectories for the family of curves y=(kx)^6.

*April 7, 2016 by TayB*

**math**

Find the orthogonal trajectories of family of the circle x2+y2=a2 where a is a parameter

*July 12, 2016 by mahesh*

**Math**

Mark each of the following True or False. ___ a. All vectors in an orthogonal basis have length 1. ___ b. A square matrix is orthogonal if its column vectors are orthogonal. ___ c. If A^T is orthogonal, then A is orthogonal. ___ d. If A is an n*n symmetric orthogonal matrix ...

*January 2, 2011 by Melissa*

**calculus**

two curves are orthogonal at a point of intersection of their tangents at that point cross at right angles. Show that the curves 2x^2+3y^2=5 and y^2=x^3 are orthogonal at (1,1) and (1,-1). Use parametric mode to draw the curves and to show the tangent lines

*November 15, 2011 by Anonymous*

**Calculus**

What is the orthogonal trajectory of y^2 - x^2 = C ??

*December 14, 2009 by Nadine*

**Math - Vectors**

Prove that vector i,j and k are mutually orthogonal using the dot product. What is actually meant by mutually orthogonal?

*August 10, 2010 by Shaila*

**calculus**

if the tangent of two intersecting circles, at their points of intersection are perpendicular, the circles are said to be orthogonal. Show that the circles x^2+y^2-6x+4y+2=0 and x^2+y^2+8x+2y-22=0 are orthogonal. find the equation of the tangent to the ellipse x^2/a^2 + y^2/b^...

*June 28, 2008 by Kelly*

**Linear Algebra, orthogonal**

The vector v lies in the subspace of R^3 and is spanned by the set B = {u1, u2}. Making use of the fact that the set B is orthogonal, express v in terms of B where, v = 1 -2 -13 B = 1 1 2 , 1 3 -1 v is a matrix and B is a set of 2 matrices

*December 13, 2010 by Kay*

**Calculus **

Resolve u=[3,4,7] into two orthogonal vectors, one of which is collinear with v=[1,2,3].

*May 23, 2010 by Arima *

**Calculus **

Resolve u=[3,4,7] into two orthogonal vectors, one of which is collinear with v=[1,2,3].

*May 23, 2010 by Arima *

**Calculus**

Find the orthoganal trajectories of the family. Use a graphing utility to graph several members of each family. y = Ce^x What am I supposed to do here? Can someone point me in the right direction?

*March 5, 2012 by Hannah*

**Math**

I'm doing a bunch of practice finals and I don't know how to approach this problem. Find a vector a such that a is orthogonal to < 1, 5, 2 > and has length equal to 6. If I want to find a vector that is orthogonal to <1,5,2>, I must take the cross product?

*December 4, 2012 by A.*

**calculus**

Find an equation of the plane orthogonal to the line (x,y,z)=(-4,-9,9)+t(-8,-1,5) which passes through the point (-9,9,-4). Give your answer in the form ax+by+cz=d.

*March 4, 2010 by zama*

**Calculus**

Determine whether the planes are parallel or orthogonal. Equations given: 5x - 3y + z = 4 x + 4y + 7z = 1 How exactly do I solve this? It wasn't covered completely in class.

*April 8, 2013 by Krista*

**Calculus**

Find an equation of the plane orthogonal to the line: (x,y,z) = (5,-6,-1) + t(3,10,-3) which passes through the point (-6,-1,5). so i got so far: x=5+3t y=-6+10t z=-1-3t Should be in form ax+by+cz+d=d Not sure where to go from there...

*September 10, 2007 by Vectors and Planes*

**calculus (vectors)**

Determine a vector that is orthogonal to the vector e = [3, -1, 4]

*October 20, 2012 by KnowsNothing*

**Calculus-PLZ help!**

Given u=3i-2j+k,v=2i-4j-3k, w=-i+2j+2k, 1 Find a unit vector normal to the plane containing v and w. 2 Find the volume of the parallelepiped formed by u, v, and w. 3 Are any of these vectors parallel? Orthogonal? Why or why not?

*May 13, 2013 by Liz*

**calculus**

I have 3 points: P(-3, 1, 2), Q(-1, 2, 3), R(2, 1, 0) and I need to find a nonzero vector orthogonal to the plane through these three points. I seem to recall this having something to do with the cross product, so I mad vectors PQ <2,1,1> & PR <5,0,-2> calculated ...

*August 24, 2011 by cross product*

**math**

Show that x^2+y^2-6x+4y+2=0 and x^2+y^2+8x+2y-22=0 are orthogonal.

*July 9, 2010 by dan*

**st joseph **

show that the circle x^2+y^2-6x+4y+2=0 and x^2+y^2+8x+2y-22=0 are orthogonal

*October 12, 2015 by kamau*

**linear algebra**

Solve for x if the vectors (2, x, 7-x) and (x, 3, -2) are orthogonal

*July 27, 2013 by Anonymous*

**Diagonalize**

construct a nondiagonal 2 x 2 matrix that is diagonalizable but not invertible. Just write down a diagonal matrix with one zero on the diagonal and then apply an orthogonal transformation. E.g. if you start with the matrix: A = [1 ,0 0,1] And take the orthogonal transformation...

*July 12, 2007 by Jeff*

**Vectors **

Determine a unit vector that is orthogonal to both u=[3,-4, 1] and v=[2,3,-4].

*May 23, 2010 by Ariza *

**Math**

Determine whether u and v are orthogonal,parallel, or neither. u=-2i+j v=3i+6j

*February 16, 2011 by Matt*

**Calc**

Determine whether u and v are orthogonal, parallel or neither u= <-4, 7>, v= <-14, 12>

*February 13, 2012 by Darius*

**calc**

If possible, find a, b, and c so that v = [a b c] is orthogonal to both w = [1 6 1] and x = [1 −9 1]

*April 2, 2015 by sara*

**Math**

Knowing u = (4,0,-3), v = (x,3,2) and that the orthogonal projection of v on u is a vector of norm 6, determine x. Thank you

*August 27, 2012 by Robert*

**math**

Determine all values of k for which each pair of vectors is orthogonal. a) (1,2) and (k,k) b) (1,2,1) and (k,2k,4)

*September 23, 2014 by Kyle*

**Math**

Determine all values of k for which each pair of vectors is orthogonal. a) (1,2) and (k,k) b) (1,2,1) and (k,2k,4)

*September 24, 2014 by Kyle*

**Algebra**

Determine all values of k for which each pair of vectors is orthogonal. a) (1,2) and (k,k) b) (1,2,1) and (k,2k,4)

*September 24, 2014 by ZZZ Top*

**mathematics**

find orthogonal trajectory of family of circles x^2+y^2+2fy+1=0.

*September 13, 2015 by Anonymous*

**Algebra**

Determine if the following two planes 2x+3y-z=4 and 3x-4y-6z=1 are parallel, orthogonal or coincidental

*September 21, 2010 by Matthew*

**Linear Algebra**

Knowing u = (4,0,-3), v = (x,3,2) and that the orthogonal projection of v on u is a vector of norm 6, determine x. Thank you

*August 26, 2012 by Robert*

**Math**

Find an equation of a plane through the point (1, 5, 1) which is orthogonal to the line x=3+5t y=5-1t z=-1+4t in which the coefficient of x is 5.

*February 1, 2010 by Derek*

**Math**

How do we know the ith of an invertible matrix B is orthogonal to the jth column of B^-1 , if i is not equal/unequal to j?

*December 8, 2010 by Nolan*

**physics**

A particle is moving in a circular trajectory because of a magnetic field. Show that regardless of the veolocity of the particle, it will take the same amount of time to complete one revolution. I'm not sure how to prove this. Should i use the formula for circle trajectories.....

*April 1, 2008 by Jessica*

**TRIGONOMETRY**

Use the dot product to determine which of the following vector pairs are orthogonal. a. v1 = (-5,5) and v2 = (1,1) b. v1 = (154,169.4) and v2 = (88,64)

*May 18, 2013 by Kate*

**Linear Algebra**

A test Question: Find the parametric equations of the line through the origin that is orthogonal to the plane 2x+4y-z=0

*December 14, 2011 by Simon*

**Geometry/Algebra/Calculus**

Why isn't the surface area of a sphere with radius r the following: 2*pi * (pi*r) That comes from the following flow of logic: Doesn't it makes sense to think of the surface area of the sphere with radius r as the the circumference of the semi-circle with radius r, pi*r (2*pi*...

*January 26, 2014 by John*

**Math**

The functional determinant of x, e^x, and e^-x is equal to___ The orthogonal trajectory of y^2 - x^2 = C is__ The slope of the normal line to y^2 = x/2 at P(1/8,1/4) is ____ help please...

*December 14, 2009 by Alvin*

**Dr. D ram D.A.V. Public school**

A parabola y²=4x cuts the circle with centre at (6,5) orthogonal then the possible points of intersection between the curves are?

*December 10, 2015 by Ankit raj*

**Neurology**

Which of the following pairs of terms identify spaces that are roughly PERPENDICULAR (orthogonal) in the human brain (give or take 30 degrees or so)?

*April 27, 2013 by Anonymous*

**mathematical physics**

Determine the scale factors for an orthogonal coordinate system (s,t,v) whose coordinates are related to the Cartesian by the following equations: x = 2st, y = s^2-t^2, z = v.

*March 22, 2015 by gourav *

**rotational mechanics**

a thin rod of length 2R and mass M is standing vertically on a perfectly smooth floor. the state of equilibrium in which the rod at rest is unstable and the rod falls. FInd the trajectories that the various points of rod describe and velocity with which the upper end of rod ...

*March 31, 2010 by pramod*

**math**

Diagonalize the given matrix and find an orthogonal matrix P such that P−1AP is diagonal -2 3 3 3 -2 3 3 3 -2

*April 23, 2015 by Anonymous*

**math**

A trigonmetric polynomial of order n is t(x) = c0 + c1 * cos x + c2 * cos 2x + ... + cn * cos nx + d1 * sin x + d2 * sin 2x + ... + dn * sin nx The output vector space of such a function has the vector basis: { 1, cos x, cos 2x, ..., cos nx, sin x, sin 2x, ..., sin nx } Use ...

*January 16, 2008 by mathstudent*

**math urgent**

let V=R^3 and S={u1,u2,u3}=[1;2;0],[1;0;0],[1;0;1] These are three vectors 1 by 3 use gschmidt to obtain an orthogonal basis and then find the coordinates of x=[1;2;3] relative to this basis.

*March 25, 2015 by kirsten*

**vectors **

how do you determine if two vectors are a) Collinear b) orthogonal?

*June 17, 2010 by Ariza *

**math**

Find an orthonormal basis for the subspace of R^3 consisting of all vectors(a, b, c) such that a+b+c = 0. The subspace is two-dimensional, so you can solve the problem by finding one vector that satisfies the equation and then by constructing another solution which is ...

*July 26, 2007 by john*

**Art History**

In linear perspective, all parallel lines converge at a(n): A. orthogonal. B. vanishing point. C. horizon line. D. picture plane. OK, my textbook says it both A and C so I am totally confused.

*November 22, 2014 by Pat*

**physics**

Two frisky grasshoppers collide in midair at the top of their respective trajectories and grab onto each other, holding tight thereafter. One is a robust 250 g beast initially moving south at 20.0 cm/s, while the other is a svelte 150 g creature initially moving north at 60.0 ...

*October 28, 2009 by physics studentt*

**math**

) Determine whether the following two planes x + 4y − z = 7 and 5x − 3y −7z = 11 are parallel, orthogonal, coincident (that is, the same) or none of these.

*April 22, 2014 by Carrolyn*

**mathematical physics**

Determine the scale factors for an orthogonal coordinate system ( s, t, v ) whose coordinates are related to the cartesion by following equations: x = 2st:, y = s^2-t^2:, z = v Also write down an expression for the square of the arc element.

*March 21, 2015 by gourav *

**mathematical physics**

Determine the scale factors for an orthogonal coordinate system (s, t, v) whose coordinates are related to the Cartesian by the following equations: x = 2st, y = s^2-t^2, z = v Also write down an expression for the square of the arc element.

*March 23, 2015 by gourav *

**linear algebra urgent**

For the orthogonal matrix A = 1/sqrt(2) -1/sqrt(2) -1/(sqrt(2)) -1/sqrt(2) verify that (Ax,Ay)=(x,y) for any vectors x and y in R2. Can someone please explain this

*April 22, 2015 by Anonymous*

**math**

can an inner product space v have a t invariant subspace U but also have an orthogonal complement that is NOT t-invariant???

*March 3, 2007 by mk*

**math**

Find the least squares approximation of x over the interval [0,1] by a polynomial of the form a + b*e^x --------------------------------------------------------- The polynomial produces an output space with two linearly independent basis vectors: u1 = 1, u2 = e^x I believe ...

*January 18, 2008 by mathstudent*

**Linear Algebra**

Find the basis for the following vector space. Please state the dimension of the vector space. S consists of all x in R3 such that x is orthogonal to n=(2,3,2)

*November 20, 2011 by Simon*

**math**

Find the orthogonal canonical reduction of the quadratic from -x^2+y^2+z^2-6xy+2yz. Also, find its principal axes,rank and signature of the quadratic form.

*March 27, 2016 by gourav bhardwaj*

**math**

Find the orthogonal canonical reduction of the quadratic from -x^2+y^2+z^2-6xy+2yz. Also, find its principal axes,rank and signature of the quadratic form.

*March 28, 2016 by gourav bhardwaj*

**physics**

An electron starting from rest acquires 4.20 keV of kinetic energy in moving from point A to point B. How much kinetic energy would a proton acquire, starting from rest at B and moving to point A? Determine the ratio of their speeds at the end of their respective trajectories.

*September 8, 2010 by VJC*

**math**

given that vectors(p+2q) and (5p-4q) are orthogonal,if vectors p and q are the unit vectors,find the dot product of vectors p and q?

*January 2, 2013 by bhawani*

**Physics**

Two projectiles are launched from ground level at the same angle above the horizontal, and both return to ground level. Projectile A has a launch speed that is twice that of projectile B. (Sketch trajectories of projectiles). Assuming that air resistance is absent, what should...

*May 24, 2010 by Christy*

**math**

given that vectors(p+2q) and (5p-4q) are orthogonal,if vectors p and q are the unit vectors,find the product of vectors p and q?

*January 1, 2013 by bhawani*

**math**

There is one step in a proof that I don't understand. Could someone please explain? u = any vector in vector space S W = finite dimensional subspace of S with orthonormal basis of vectors {v1, v2, ..., vn} The theorem to prove is: u can be expressed exactly one way as u = w1...

*January 5, 2007 by mathstudent*

**Math**

Show that A = [3 2 4 2 0 2 4 2 3] is distinguishable even though one eigenvector has algebraic multiplicity 2. Do this by brute force computation. Why would you expect this to be true, even without calculation? Then, for the A, write A= Q lambda Q^(T) where Q's columns are ...

*July 12, 2012 by Christopher*

**Math**

1. P5 is an innerproduct space with an inner product. We applied the Gram Schmidt process tot he basis {1,x,x^2,x^3,x^4} and obtained the following as the result {f1,f2,f3,f4,x^4+2}. a. What is the orthogonal complement of P3 in P5 with erspect to this inner product? b. What ...

*April 13, 2012 by Scott*

**linear algebra-urgent**

1)let w=[3;4] and u=[1;2] a) find the projection p of u onto w. I found this to be p=[1.32;1.76] b) find a scalar k for which the vector kp has a norm that is equal to one. k=? d)find a vector that is orthogonal to span{w} s=? how do I do this

*April 8, 2015 by samantha*

**math**

Prove that if A is a symmetric n x n matrix, then A has a set of n orthonormal eigenvectors. http://ltcconline.net/greenl/courses/203/MatrixOnVectors/symmetricMatrices.htm I've read the entire page and while it's on the correct topic, it doesn't prove what I'm looking to prove...

*January 19, 2007 by mathstudent*

**Linear Algebra**

Ok this is the last one I promise! It's from a sample exam and I'm practicing for my finals :) Verify if the following 4 points are consecutive vertices of a parallelogram: A(1,-1,1); B(3,0,2);C(2,3,4);D(0,2,3) (b) Find an orthogonal vector to the plane containing the ...

*August 27, 2012 by Robert*

**Math**

cotA = tan(90degrees- A) True or False The vectors <4, 5> and <-10, 8> are orthogonal True or False 2<3, 5> = 16 True or False <2. 3>*<4, 3> = <2, 6> + <4, 5> = Find c if a = 3, b = 5, and A = 30degrees=

*December 3, 2015 by April*

**Math**

cotA = tan(90degrees- A) True or False The vectors <4, 5> and <-10, 8> are orthogonal True or False 2<3, 5> = 16 True or False <2. 3>*<4, 3> = <2, 6> + <4, 5> = Find c if a = 3, b = 5, and A = 30degrees=

*December 3, 2015 by April*

**ART HELP!**

Please help me with this: What step did the artist clearly use when drawing a two-point perspective image? A. A vanishing point *** B. Only orthogonal lines C. Imagery in the backgrounds larger than in the foreground. D. Imagery in the foreground lighter than in the background.

*October 15, 2015 by HelpME:(!!!*

**ART! URGENT**

What shold an artist do first in order to draw a two-point perspective image? A. Use orthogonal lines. B. Use value changes C. Draw all vertical lines D. Draw a horisan line C?

*October 6, 2014 by Anonymous*

**math**

The runway of an airfield faces west. An airplane, flying in the direction of north-east at a height of 2km and at a speed of 400km/h, on a path which passes over a point 3km west of the runway end, is spotted sqrt(29)km horizontally(south-western quadrant) from the runway end...

*November 20, 2011 by Jainee*

**Art**

1. Which step did the artist clearly use when drawing this image? two vanishing points only orthogonal lines imagery in the background larger than in the foreground imagery in the foreground lighter than in the background 2. Because of the use of value change in this image, ...

*September 15, 2016 by PolkaDot*

**linear algebra**

Find the inverse of each of the following orthogonal matrices. A= [1 0 0 0 cos(theta) sin(theta) 0 -sin(theta) cos(theta)]

*April 21, 2015 by kat*

**physics**

Two bar magnets (1) and(2) orthogonal to each other create at point M (intersection of their supports) the respective magnetic field vectors B1 and B2 of intensity: B1=o.oo3T and B2=0.004T a)Determine the names of the poles of the magnets. b)Construct the resultant field ...

*May 17, 2016 by angy*

**Pre-Calculus/Calculus**

I am too embarassed to ask this Calculus (really pre-calculus) question in tutoring, because I know I should know. Is the inverse of f(x)=3x-1 actually f(x)=1/3x+1? How do I find it? What if it asks the same equation replaced with f to the -1 power (x)? I think I know how the ...

*July 1, 2008 by Molly*

**Trigonometry**

Which of the following pairs of vectors are orthogonal? (i) v = 18i − 3j and w = −i − 6j (ii) v = 15i − 2j and w = −4i + 30j (iii) v = 3i − j and w = i + 3j A. (i) and (ii) B. (i) and (iii) C. (ii) only D. (i) only 17.

*August 1, 2012 by Anonymous*

**Finite Math**

Fifty percent of students enrolled in calculus class have previously taken pre-calculus. Thirty percent of these students received an A for the calculus class, whereas twenty percent of the other students received an A for calculus. Find the probability that a student selected...

*June 10, 2015 by Nikole*

**Calc 3**

Consider the points below. P(2, 0, 2), Q(−2, 1, 4), R(7, 2, 6) (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R. (b) Find the area of the triangle PQR. * I answered part a and got the correct answer with the vector <0,26,-13> I found...

*September 8, 2012 by Backy*

**Calculus**

What is the use of Calculus? How is it use in jobs? What jobs use Calculus? Calculus is used in engineering, economics, any physical science, and in business (e.g., actuary studies and statistics).

*May 5, 2007 by Anonymous*

**linear algebra**

3. Suppose A is symmetric positive definite and Q is an orthogonal matrix (square with orthonormal columns). True or false (with a reason or counterexample)? a) (Q^(T))AQ is a diagonal matrix b) (Q^(T))AQ is a symmetric positive definite matrix c) (Q^(T))AQ has the same ...

*December 1, 2010 by michael*

**linear math**

Let u=(2,0,k,-1) v=(-4,0,-3,2) and w= (0,1,0,0). Each answer must be justified(there is no answer where k=nothing) a) find all values of k, (if any) for which u is orthogonal to v b) Find all values of k (if any) for which the set {u,v,w} is linearly independent. Find all ...

*November 20, 2011 by Simon*

**math**

Can I please get my answers double checked? True or False 1. The tangent function is used to find the angel between two vectors. True 2. If two vectors are orthogonal, then their dot product is -1. False 3. The length of the projected vector is always shorter than the length ...

*June 17, 2013 by mysterychicken*

**Biology**

Theme A: Surface anatomy of the brain Of the following pairs of directional terms, which pair contains terms that define PERPENDICULAR (orthogonal) directions when applied to the identified region of the central nervous system? in the brainstem, ventral & anterior in the ...

*April 10, 2013 by Anonym*

**Calculus**

Hellp I have a calculus test on Monday 2. Calculate the slope of the tangent to the given function at the given point or value of x a. f(x)=3/x+1,P(2,1) b.h(x)=2/squareroot x + 5, P(4,2/3) ****Full solutions to please because I do not know what I'm doing and so confused with ...

*February 27, 2016 by Morris*

**Maths trigonometry **

using basic trigonometry derive the following equations, which are used to convert cartesian to polar coordinates and vice versa. why is the cos(phi)in the x term? x = R cos (phi) sin (lamda) y = R sin (phi) they are used in orthogonal projection of a spherical surface to a ...

*October 16, 2011 by forest*

**Algebra**

In an interview of 50 math majors, 12 liked calculus and geometry 18 liked calculus but not algebra 4 liked calculus, algebra and geometry 25 liked calculus 15 liked geometry 10 liked algebra but neither calculus nor geometry 2 liked geometry and algebra but not calculus. Of ...

*January 11, 2010 by Anita*

**art history**

1. Art enabled new understandings in this field 2. Credited with being the founder of Renaissance painting in the Netherlands 3. his most famous painting depicts sin and redemptionhe 4. brought Germany into the mainstream of Renaissance art 5. objects appear to get smaller ...

*June 23, 2014 by bekah*

**Calculus**

What is the simplest solution to the Brachistochrone problem and the Tautochrone problem involving calculus? (I know that the cycloid is the solution but I need a simple calculus proof as to why this is the case)

*May 27, 2010 by Sam*

**math**

8. In an interview of 50 math majors, 12 liked calculus and geometry 18 liked calculus but not algebra 4 liked calculus, algebra, and geometry 25 liked calculus 15 liked geometry 10 liked algebra but neither calculus nor geometry 2 liked geometry and algebra but not calculus. ...

*November 30, 2009 by poo*

**physics**

The drawing shows a steel ball rolling along a flat surface at constant velocity from point A to point B. A magnet exerts an attractive force on the ball only for the short time that passes as the ball rolls past point B. Which one of the following trajectories does the ball ...

*June 8, 2014 by Sam*

**physics**

he drawing shows a steel ball rolling along a flat surface at constant velocity from point A to point B. A magnet exerts an attractive force on the ball only for the short time that passes as the ball rolls past point B. Which one of the following trajectories does the ball ...

*June 8, 2014 by Sam*

**Art Please check answers**

1. These two technological advancements of the Industrial Revolution helped lead to the development of Impressionism. the railroads and the printing press the railroads and photography photography and electric lighting** the automobile and electric lighting 2. According to ...

*December 3, 2012 by Anonymous*

**Art Please check answers**

1. These two technological advancements of the Industrial Revolution helped lead to the development of Impressionism. the railroads and the printing press the railroads and photography photography and electric lighting** the automobile and electric lighting 2. According to ...

*June 2, 2013 by Mstv1112*

**Physics (help steve)**

Exact solutions for many-body problems are rarely encountered in physics. The following problem deals with a non-trivial motion of four charges. Due to the symmetry of the problem it is possible to determine the trajectories of the charges analytically. Four identical ...

*July 9, 2013 by Kay*

**SCIENCE,PHYSICS**

Exact solutions for many-body problems are rarely encountered in physics. The following problem deals with a non-trivial motion of four charges. Due to the symmetry of the problem it is possible to determine the trajectories of the charges analytically. Four identical ...

*July 10, 2013 by GT*