Symmetric

Question: Find a symmetric equation of the line which is orthogonal to both L1 and L2 and passes thru their intersection? L1: x=1-4t, y=3t+2, z=4-2t L2: x=2-s, y=1+s, z=6s-2 Find a symmetric equation of the line which is orthogonal to both L1 and L2 and passes thru their point of intersection.

Answer: mizzou calc 3 sucks

Question: What is a symmetric point in regards to quadratic equations? Reviewing for a test tomorrow, the study guide says to be able to "define a symmetric point." What is this, exactly? Just the point in the middle of the parabola when it crosses the X-axis?

Answer: turning point or vertex

Question: What is the difference between symmetric and antisymmetric relational sets? Symmetric means it contains every element in the set?

Answer: normally a relation R on a set A (which can be viewed as a subset of AxA) is called symmetric if xRy --> yRx, or equivalently, if (x,y) in R ---> (y,x) in R. an anti-symmetric relation is one for which: xRy and x ≠ y --> ⌐(yRx) or, again, if (x,y) is in R, and x ≠ y, (y,x) is not in R. sometimes this is expressed like this: xRy and yRx --> x = y, or: if (x,y) is in R, and (y,x) is in R, x = y. equality and more generally equivalence relations, are examples of symmetric relations, ≤ is an example of an anti-symmetric relation. if you view a relation on A as a subset of A x A, symmetric relations are symmetric about the diagonal of A: ΔA = {(x,x) in A x A}.

Question: Is being symmetric to the origin the same as being symmetric to the x axis? Is being symmetric to the origin the same as being symmetric to the x axis? What would some examples be?

Answer: no, a function cannot be symmetric to the x axis, as that would mean that there are two range values for one x value. (2 y's for one x) an example of a function symmetric about the origin is x^3, x^5, cosx

Question: How to find symmetric equations for the line of intersection of two planes? I am given the planes z = 3x - y - 7 and z = 4x+2y-6, and the symmetric equation z/10. How am I supposed to find this line?

Answer: use perhaps the Riemann's dzeta function

Question: What is the center of a symmetric group? How can you show that the center of a Symmetric group (Sn) consists of only the identity, given n>2. Does the fact that any permutation can be represented as the product of 2-cycles play a part here? I'm truly stuck! Please help! This is a problem from modern/abstract algebra.

Answer: Suppose z is in Sn and that z is not the identity. Then z must take some element a to some different element b. Let c be any element of {1, ..., n} different from both a and b (we can find one since n > 2). Then the cycle g = (b c) does not commute with z, since gz(a) = g(b) = c but zg(a) = z(a) = b. Hence no non-identity element of Sn can be in the centre of Sn, so the centre of Sn is just the identity.

Question: What is a relation on the real number's which is symmetric and transitive, but not reflexive? I cannot seem to find any. I've found plenty that are neither reflexive nor transitive ("is not equal to", "is greater than or less than" etc.) but cannot find any relations which are symmetric and transitive, but not reflexive. Ideas?

Answer: Define a relation R declaring a R b if ab > 0. We readily see R is symmetric and transitive. But it's not reflexive, because 0 is not in the relation R with 0. 0 is the only real number that is not in the relation R with itself. Actually, 0 is not in the relation R with any real number. Since a R a doesn't hold for every real number a, R is not reflexive. Observe that, if R is symmetric and transitive and, for some a, there is b such that a R b, then b R a and, by transitivity, a R a. So, if R is symmetric and transitive but not reflexive, then there must exist some a in the set that is not in the relation R with any element of the set.

Question: What is the symmetric point to this quadratic graph? OK, the equation of the graph is y-73/8= -2(x+7/4)^2, the vertex is (-7/4, 73/8), the axis of symmetry is -7/4, and the y-intercept is (0,3) What is the symmetric point?

Answer: the symmetric point... hmmm you actually have an axis of symmetry... the intersection of the axis and the curve is the vertex. this might be the point you need... §

Question: What is the most symmetric arrangement of points in 3-D space so that they are all equidistant? By symmetric arrangement, what I mean is that when the points are viewed in x-y or x-z or y-z planes, they look the same in the 3 planes. Imagine placing small marbles in space. We want each neighboring marble to be equidistant. And when we view those marbles, they appear to have a symmetric pattern. Points placed on vertices of a cube are the most symmetrical arrangement since they look alike in whatever plane you look. However all neighboring points are not equidistant. They will all be either x apart or sqrt(2)x apart. We want all points to be equidistant. Dutch_prof has given a v detailed answer and i suppose i should give 10 pts---but I disagree with the answer. Because a regular tetrahedron is not 100% space filling (3-D space). When stacked the arrangement of tetrahedrons will turn out to be assymetric because of the fact that tetrahedrons can not be completely stacked without overlapping or gaps--IMHO. Correct me if I am wrong...

Answer: If you want all points to be equidistant, the largest arrangement you can possibly make is a regular tetrahedron, with four points. Now you want a proof :) Let the first point be P1 = (0,0,-1) and the second point be P2 = (0,0,1). [The coordinate system can always be translated and scaled to make this happen.] The distant between the points is 2. Any additional points P3 should also have distant 1 to each of the given points. All points with this property lie on a circle with radius sqrt 3 around the origin, in the xy-plane. Proof: because d(P1,P3) = d(P2,P3) = 2 x^2 + y^2 + (z+1)^2 = 4 ==> x^2 + y^2 + z^2 + 2z = 3 x^2 + y^2 + (z-1)^2 = 4 ==> x^2 + y^2 + z^2 - 2z = 3 It follows immediately that 2z = -2z, so z = 0. The resulting formula is x^2 + y^2 = 3, which describes the circle. Let's pick the point P3 on the circle. By rotating the coordinate system around the z-axis we can always make this point P3 (sqrt 3, 0, 0). A fourth point P4 must not only lie on the circle (x^2 + y^2 = 3), but also have distance 2 to point P3. This gives (x - sqrt 3)^2 + y^2 + z^2 = 4 x^2 + y^2 + z^2 - (2 sqrt 3) x = 1 Using z = 0 and x^2 + y^2 = 3, we get -(2 sqrt 3) x = -2 x = 1/sqrt 3 (1/sqrt 3)^2 + y^2 = 3 --> y^2 = 8/3 --> y = +/- 2/3 sqrt 6 For point P4 we have two choice: (1/sqrt 3, +/- 2/3 sqrt 6, 0). However, only one point can be added to our set because the distance between the two possible points is 4/3 sqrt 6, which is not equal to 2.

Question: A symmetric die is rolled 3 times. If it is known that face 1 appeared at least once what is the probability? A symmetric die is rolled 3 times. If it is known that face 1 appeared at least once what is the probability that it appeared exactly once. Thanks!

Answer: There are 3*5*5 = 75 outcomes in which 1 appears exactly once, and 6*6*6 - 5*5*5 = 91 outcomes in which 1 appears at least once, so the conditional probability is 75/91.

Question: How is f(x) = 1-|x| symmetric about the origin on the domain -1<x<1? I think that it's only symmetric with respect to the y axis and not with respect to the origin, but the answer key says with respect to the origin. How is this true?

Answer: Wrong category. This question should be in the Math category.

Question: How does a symmetric weight-carrying bar in the gym maintain balance even if we load only one side of the bar. Just to clarify. I am talking about the bar that you load up with circular weights at either ends to do chest press. So many times i have just loaded one side (either one) with upto 15-20 kgs and still the rod does not tip over!! Since the rod seems symmetric, i am having difficulty explaining this.

Answer: It has to do with the center of gravity of the bar and some simple number crunching. Presumably the bar is still resting on the rack when you start to load up one side. This means that the point of contact between the bar and the rack closest to the side you're loading with weights becomes the fulcrum for a lever you're setting up by overloading one side. My best guess is that you're just not putting enough weight on. In my experience, those bars weigh upwards of 25 lbs (Olympic bars weigh 30 lbs), or 14-15 kgs. When you set up the lever system, the majority of the weight of the bar is acting as a counterweight. It is also true in lever systems that the distance from the fulcrum effects the balance. The weights you're loading up are much closer to the fulcrum than the center of gravity of the other side of the bar which is acting as the counterbalance. It's a pretty simple operation: Mass * distance= rotational force (i.e the tendency for the bar to flip). So, for the sake of example, let's say that the bar weighs 15 kg and is also 15m long(obviously a gross exaggeration, but it makes the math very simple), and that when placed on the rack, 2kg worth of bar stick out on each end for you to load up the weights. This means when you start loading up the weights, you have 2kg of bar + the weights themselves multiplied times a distance of 1m (assuming that the weights added are distributed uniformly across the portion of the bar you have to work with). On the other side, you have 13kg of bar with a center of gravity 6.5m away from the pivot point. This means to flip the bar over, you would need about 85kg of weight! In short, there's nothing special about the bar, it's just the weight distribution and the properties of lever systems.

Question: What is the difference between positive-definite matrix and a symmetric banded matrix? are they both same? can a symmetric banded matrix be the positive-definite matrix as a special case? can you suggest an example? ty

Answer: Matrices used in statistics The following matrices find their main application in statistics and probability theory. * Bernoulli matrix — a square matrix with entries +1, −1, with equal probability of each. * Centering matrix — a matrix which, when multiplied with a vector, has the same effect as subtracting the mean of the components of the vector from every component. * Correlation matrix — a symmetric n×n matrix, formed by the pairwise correlation coefficients of several random variables. * Covariance matrix — a symmetric n×n matrix, formed by the pairwise covariances of several random variables. Sometimes called a dispersion matrix. * Dispersion matrix — another name for a covariance matrix. * Doubly stochastic matrix — a non-negative matrix such that each row and each column sums to 1 (thus the matrix is both left stochastic and right stochastic) * Fisher information matrix — a matrix representing the variance of the partial derivative, with respect to a parameter, of the log of the likelihood function of a random variable. * Hat matrix - a square matrix used in statistics to relate fitted values to observed values. * Precision matrix — a symmetric n×n matrix, formed by inverting the covariance matrix. Also called the information matrix. * Stochastic matrix — a non-negative matrix describing a stochastic process. The sum of entries of any row is one. * Transition matrix — a matrix representing the probabilities of conditions changing from one state to another in a Markov chain

Question: How much can I perturb a symmetric positive definite matrix? I have a matrix K which is symmetric positive definite. I need to normalize the columns of K, that is, Kij = Kij / sum_[i] Kij. This is the same as doing K = KN, where N is a diagonal matrix with entries Nii = 1/sum_[i] Kij. My question is: which matrices K will have a corresponding KN which is still positive definite?

Answer: A real symmetric matrix M is positive definite if and only if all the following matrices have a positive determinant (the Sylvester criterion): the upper left 1-by-1 corner of M the upper left 2-by-2 corner of M the upper left 3-by-3 corner of M ... M itself Here, since K is positive definite, the following matrices have a positive determinant: the upper left 1-by-1 corner of K the upper left 2-by-2 corner of K the upper left 3-by-3 corner of K ... K itself Consider L = KN where N is a diagonal matrix. Then det[L] = det[K]*det[N] and it is easy to show that det[upper left i-by-i corner of L] = det[upper left i-by-i corner of K] *det[upper left i-by-i corner of N] for i = 1,2, ... Since det[upper left i-by-i corner of K] > 0 The matrix L is positive definite if and only if det[upper left i-by-i corner of N] > 0 i.e., since N is diagonal N1>0, N1*N2>0, ... i.e., all Ni > 0 In your case Kij = Kij / sum_[i] Kij is positive definite if and only if sum_[i] Kij for all i

Question: How do symmetric airfoils generate lift? Without camber, it would seem that pressure on either side of the airfoil would be equal regardless of velocity and thus there would be no lift generated. Are symmetric (no-camber) airfoils dependent on positioning (positive angle of attack on level ground?

Answer: A symmetric airfoil generates lift based on angle of attack. (The angle the airfoil meets the oncoming air) Zero angle of attack = Zero lift.

Question: Help with symmetric equations and the angle between planes? 1) Find symmetric equations for the line of intersection of the planes x+y-z=2 and 3x-4y+5z=6 2) Find the angle between these planes (Round answer to nearest tenth of a degree). Can anyone offer some guidance? Any help would be greatly appreciated, thanks!

Answer: The solution is here http://www.math.ucla.edu/~ronmiech/Calcu…

Question: How do symmetric airfoils lift off? I know that you can create lift using either angle of attack or camber in an airfoil but how can a symmetric airfoil lift off? There won't be any angle of attack since the plane is on the ground and there is no camber since it is symmetric, so where does all the lift come from? Is it the ground effect?

Answer: Angle of attack is changed by the elevator. Pull back on the stick makes the tail of the plane go down and increases the AOA. It works regardless of the type of wing. If you left the elevator in the neutral position on take off, the tail wheel will come off the ground until there's zero AOA and the plane will just roll down the runway no matter how fast it goes. You have to pull back on the stick to increase AOA to get the plane off the ground.

Question: Is a probability distribution curve always bell-shaped and symmetric about the mean? Is a probability distribution curve always bell-shaped and symmetric about the mean?

Answer: Nope. Not even close. There are at least a dozen different continuous probability distribution curves, and probably a dozen more discrete probability distribution curves.

Question: What is wrong with this "proof" that a symmetric, transitive relation R is also reflexive? What is wrong with the following "proof" that a symmetric, transitive relation R is also reflexive? Suppose xRy Then yRx by symmetry. xRy and yRx implies xRx by transitivity. Thus R is reflexive. Can you give an example of a relation that is symmetric and transitive but not reflexive?

Answer: Is there any y such that xRy? Maybe there no such y. The exemple is R = {(1,2),(2,1),(1,1),(2,2)} on a set A={1,2,3}

Question: How can I show that all symmetric 2x2 matrices form a subspace of the vector space M2 of all square 2x2..? matrices? I also need to figure out the basis for this subspace. Also what does it mean when I need to solve for the same concept but in skew-symmetric matrix?

Answer: To show the property of being a subspace you need only show that it's closed under scalar multiplication and addition of vectors. Suppose a symmetric 2x2 matrix M. [a b] [b c] Suppose we take a scalar from the field the vector space is over called s. Then s*M is equivalent to: [s*a s*b] [s*b s*c] Clearly this matrix is also symmetric, so our first condition is satisfied. Suppose another matrix M' defined as follows: [d e] [e f] Then M+M' is in the space if the sum is symmetric. [a+d b+e] [b+e c+f] which is clearly a diagonal matrix. This demonstrates that the set of symmetric 2x2 matrices is a subspace of the set of all 2x2 matrices. As for your question about skew symmetric, apply the same methodology to skew symmetric matrices and see if the result is true. A skew symmetric matrix is one in which the transpose of the matrix is the same as its negative. In other words, it's a symmetric matrix in which the signs of the elements flipped across the diagonal are also flipped. For instance: [2 3] [-3 4] is skew symmetric.