Product Definition and 1000 Threads

Definition/Summary The cross product of two vectors \mathbf{A} and \mathbf{B} is a third vector (strictly, a pseudovector or axial vector) \mathbf{A}\times\mathbf{B} perpendicular to both of the original vectors, with magnitude equal to the product of their magnitudes times the (positive)...

Homework Statement Hi, I am having difficulty with the following proof: Let V be an inner product space (real of dimension n) with two inner products in V, <,> and [,]. Prove that there exists a linear mapping on V such that [L(x),L(y)] = <x,y> for all x,y in V. I am stuck as to where...

Homework Statement Youtube: Lec 2 | MIT 18.02 Multivariable Calculus, Fall 2007 (Video time frame: between 11:00 minutes and 12:30 minutes) Find the area of a triangle. Area = \frac{1}{2}(base)(height) = \frac{1}{2}|a||b|sinθ The lecturer says to first find cosine of the angle using dot...

I understand that dot product of vectors means projecting one vector on to the other. But I don't understand what is the physical significance of a cross product? I have read that cross product gives the area of the parallelogram which has each of the vectors as its sides...but why do we want to...

Why is Ksp not defined for soluble salts? Also, when an equilibrium is established between the solid, undissolved salt and the ions in the saturated solution, won't adding more solid shift the equilibrium to the left causing more ions to form?

Hello, Been a long time lurker, but first time poster. I hope I can be very thorough and descriptive. So, I have been battling with a double inner product of a 2nd order tensor with a 4th order one. So my question is: How do we expand (using tensor properties) a double dot product of the...

Hi,let: 0->A-> B -> 0 ; A,B Z-modules, be a short exact sequence. It follows A is isomorphic with B. . We have that tensor product is right-exact , so that, for a ring R: 0-> A(x)R-> B(x)R ->0 is also exact. STILL: are A(x)R , B(x)R isomorphic? I suspect no, if R has torsion. Anyone...

The definition (taken from Robert Gilmore's: Lie groups, Lie algebras, and some of their applications): We have two vector spaces V_1 and V_2 with bases \{e_i\} and \{f_i\}. A basis for the direct product space V_1\otimes V_2 can be taken as \{e_i\otimes f_j\}. So an element w of this space...

In Nakahara's book, the interior product is defined like this : i_{x} \omega = \frac{1}{r!} \sum\limits_{s=1}^r X^{\mu_{s}} \omega_{\mu_{1}...\mu_{s}...\mu_{r}}(-1)^{s-1}dx^{\mu_{1}} \wedge ...\wedge dx^{u_{s}} \wedge...\wedge dx^{\mu_{r}} Can someone give me please a concret example of...

Homework Statement when compund P ( CH2=CHCH2CH3) and Q ( CH3CH=CHCH3) is reacted with steam , compound T which is optically active formed. draw the structure of T . the ans is on the left. my ans is on the right. is my ans ( on the right) accepted , why and why not? Homework Equations...

5g of AuBr3 (Ksp = 4.0 x 10^-36) are placed in 25 ml of water, how many grams of Au ions are dissolved in the 25 ml? My instructor used the conversion factor (6.2 x 10^-10 mol Au ions) to get from 25 ml H2O to grams Au. I believe the conversion from ml of H2O to grams of Au is: 25ml H2O x...

Hi everyone, I have to find out how to do cross and dot product for two vectors in non-orthogonal coordinate system. thanks

So, plenty of us have probably heard about the recent hot car death that will probably become a hugely controversial media circus that everyone will forget about in a few months. I don't know whether the death was intentional or not, but I hope the court makes the correct decision either way. I...

Homework Statement Prove that the product of four consecutive integers is always one less than a perfect square. The Attempt at a Solution I tried looking at the product (n-1)(n)(n+1)(n+2)=x^2-1 but i couldn't seem to get anything useful out of it. I added one to both sides . I tried...

1. Here is the prompt: http://imgur.com/mfbPidG 2. F = qv x B 3. At first this seemed like a simple cross product problem, and it probably still is, but I'm really confused as to what "3.70E6 m/s/ in the (i+j+k)/sqrt(3) direction" means, so I don't know how to set up my problem anymore. Could...

Homework Statement If u(t) = σ(t) . [σ'(t) x σ''(t)], show that u'(t) = σ(t) . [σ'(t) x σ'''(t)]. Homework Equations The rules for differentiating dot products and cross products, respectively, are: d/dt f(t) . g(t) = f'(t) . g(t) + f(t) . g'(t) d/dt f(t) x g(t) = f'(t) x g(t) +...

Homework Statement In this problem, you will write code that computes the Kronecker product of two arrays. Suppose A is a numeric array of size r-by-c and B is a numeric array of size n-by-m. Then the Kronecker product of A with B is a numeric array, of dimension rn-by-cm, defined as:Homework...

I was reading in my textbook that the scalar product of two complex vectors is also complex (I assuming this is true in general, but not in every case). However for the general definition (the inner product), each element of one of the vectors needs to be its complex conjugate. I learned this...

Here is the claim I am trying to prove: Suppose we have two vectors \mathbf{r} and \mathbf{s}. I would like to show that there are only two directions in which the resultant vector of the cross product \mathbf{r} \times \mathbf{s} can point, parallel and antiparallel. How might one prove...

Hi, I have an exercise whose solution seems too simple; please double-check my work: We have a product manifold MxN, and want to show that if w is a k-form in M and w' is a k-form in N, then ##(w \bigoplus w')(X,Y)## , for vector fields X,Y in M,N respectively, is a k-form in MxN. I am...

Homework Statement Calculate the value of y in the expression below: 10y = 103.2 × 102.4 × 10-1.8 × 1000.3 × 100-0.5 Homework Equations The Attempt at a Solution 10y= 103.8*100-0.2 Don't know how to move on from here as have no other examples like it? I am thinking I can use logs, however...

Hey! :o We know that: $$(x,x)=0 \Rightarrow x=0$$ When we have $\displaystyle{(x,y)=0}$, do we conclude that $\displaystyle{x=0 \text{ AND } y=0}$. Or is this wrong? (Wondering)

Is the trace of an outer product of a normalized state eq. (psi) equal to 1? Thanks, Chris Maness

1."In this exercise, you will be finding the resultant torque from the cross product of a lever arm with a force vector. The lever arm vector is A=2.0i+3.0j. The force vector is B=3.0i-4.0j. Find A x B B x A and 2A x 3B 2.My teacher has been sick the past few days so hasnt taught us...

Homework Statement In an equilateral triangle with sides u, v, w, where they are all unit vectors, find u dot w. Homework Equations u dot w = |u||w|cosθ The Attempt at a Solution The answer is ##\frac {-1} {2} ## cos(120) = -1/2 Elsewhere, I read the statement that since these are...

Homework Statement I have been sitting here for the last hour trying to figure it out but I can't seem to be able to find what I'm doing wrong. I need to expand an equation. Homework Equations a2 - a - 3 The Attempt at a Solution a2 - 1a - 3 The product is -3 and the sum -1...

If the discrete summation is symbolized by ##\sum## and the continuous by ##\int##, so, by analogoy, the discrete product is symbolized by ##\prod## and you already thought what is the symbol for the continuous product?

I'm taking "physics for engineers" right now - the condensed 4 hour summer course over 7 weeks. I'm doing fine in the class. I feel confident about the ideas and concepts we've covered so far, sure enough, but I'm having a hard time grasping the concept (geometrically at least) of the...

Hi, i was wondering how the following expression can be decomposed: Let A=B°C, where B, C are rectangular random matrices and (°) denotes Hadamard product sign. Also, let (.) (.)H denote Hermitian transposition. Then, AH *A how can be decomposed in terms of B and C ?? For example, AH...

Hi, I'm reading a book about fluid dynamics and I found some strange product between tensors. It's written like this: v=S:∇I , where S and I are matrices and v is a vector. Symbol : usually denotes Frobenius inner product. In this case we have a product of a matrix with a tensor of rank 3 and...

If a system is made up by two subsystems, for example, the atom and the photon. and let's assume the state of the atoms is described by |\phi\rangle, while the state of the photons can be described by |n\rangle, The Kronecker product of the |\phi\rangle and |n\rangle can be used to describe the...

I know intuitively that the Cross Product of two vectors ##\vec{A}## and ##\vec{B}## represents another vector ##\vec{A \times B}## perpendicular to it. In study of physics we come across this situation a lot. Hence I can visualize some applications of it I know that the dot product of...

suppose you have two random variables X and Y which are independent, we want to form a new random variable Z=XY, if f(x) and f(y) are density functions of X and Y respectively what is the density function of Z? I tried taking logs and applying convolution, but it did not really work

Homework Statement Hi all, I'm having trouble evaluating the product g_{αβ}ϵ^{αβγδ}. Where the first term is the metric tensor and the second is the Levi-Civita pseudotensor. I know that it evaluates to 0, but I'm not sure how to arrive at that. The Attempt at a Solution My first thought...

I have: dVμ = (∂Vμ/∂xη)dxη where Vμ is a contravariant vector field I believe the () term on the RHS is a covariant tensor. Is the dot product of () and dxη a scalar and how do I write this is compact form. I know how this works for scalars but am not clear when tensors are involved.

Hi there! Im new to this forum so pardon me if this is the wrong place to post my question. Basically, I have an idea with regards to doing some modification to phone screens. However, there is no such technology (yet) for this and I have zero knowledge and background pertaining to...

Hello I am trying to learn something and really hope to get some help.. i know the use of envelope detector.. but when to use product detector? i know product detector will be more expensive than envelope detector.. please help

Vectors a and b correspond to the vectors from the origin to the points A with co-ordinates (3,4,0) and B with co-ordinates (α,4, 2) respectively. Find a value of α that makes the scalar product a\cdotb equal to zero, and explain the physical significance. My attempt: The scalar product...

Cross product is used to find the perpendicular vector of two vectors. If there is any two vectors in a plane then there is always a perpendicular vector to both of them. So in circular motion if the motion is horizontal then is there a perpendicular force to the object in circular motion?

Hey guys, just trying to understand how the quotient rule is derived, so I head over to wikipedia and saw this: But I'm having some difficulty understanding what goes on between these two steps: Could someone shed some light on this?

I asked this in the logic&probability subforum, but I thought I'd try my luck here. ... Let (A,\mathcal A), (B,\mathcal B) be measurable spaces. Let p be a probability measure on (A,\mathcal A), and let q:A\to\mathcal P(B,\mathcal B) be a measurable function which takes each a\in A to...

Homework Statement I am trying to calculate the flux for the octant of a sphere, and I am trying to figure out how the mathematics, dot products, and dA works in the integral. I already did the quadrant for \hat{θ} where θ= π/2 (the bottom quadrant) and I did the left quadrant where \hat{n}...

Vectors -- dot product and cross product? Hello may i know when to dot product and cross product?? both look to same to me..

Let (A,\mathcal A), (B,\mathcal B) be measurable spaces. Let p be a probability measure on (A,\mathcal A), and let q:A\to\mathcal P(B,\mathcal B) be a measurable function which takes each a\in A to some probability measure q(\cdot|a) on (B,\mathcal B). Then there is a unique probability...

Let's say A and B are 2 vectors with length in cm and the angle between them is 170°. Obviously, the dot product of A and B will give cm2 as unit but since the value of cos(170) is negative, will the dot product be negative (something)cm2?

I tryied make the convolution product between x² and x³ by ##\int_{- \infty}^{+ \infty} f(u) g(x-u) du## and the result is an indeterminate form, however, by defintion ##\int_{0}^{x} f(u) g(x-u) du##, the result is 1/60 x6. So, \int_{- \infty}^{+ \infty} f(u) g(x-u) du \overset{?}{=}...

A spin 1/2 particle A undergoes decay A→B+C+D Where it is known that B and C are also spin 1/2. The complete set of allowed values of spin of D It was a Multiple Choice Question and options given were 1) 1/2,1,3/2,2,5/2,3,... 2) 0,1 3) 1/2 only 4) 1/2,3/2,5/2,7/2,... I tried the...

Hi all. I'm having trouble understanding the cartesian product of a (possible infinite) family of sets. Lets say \mathcal{F} = \{A_i\}_{i \in I} is a family of sets. According to wikipedia, the cartesian product of this family is the set \prod_{i \in I} A_i = \{ f : I \to \bigcup_{i...

I am trying to understand the difference from a physical phenomena point of view, not just math. Surprisingly I think I got the cross product like in rotational momentum. You have the momentum vector and we have effective distance from the momentum vector R that needs to be perpendicular to the...

Dear All, I am unable to understand a simple mathematics relation. I spent 2-3 hours to Google multi-variable mathematics and have studied some concepts, still i am missing/confusing some basics. The problem I have at hand is following. Vector p can be written as p = (p1, p2, p3) = n(sin θ3...