Product Definition and 1000 Threads

I'm just trying to understand from a linear algebra standpoint how they define dot product from the inner product and how this gives rise to a definition of length and angle. somehow there is a way to combine points in space to a scalar value that unambiguously determines length and angle? Is...

Homework Statement Hello, I've been trying to solve this problem, but in the examples that my teacher gave me didn't include something like this, I know how to calculate area but only if I have all the coordinates established. I need to find the area using the cross product. Homework...

Homework Statement http://postimg.org/image/lgphyvggz/ Homework Equations The Attempt at a Solution can someone explain where that transpose came from in (3.3)?

when differentiating e^(at) * (cos(bt) + isin(bt)) are you able to use product rule to find the derivative considering (cos(bt) + sin(bt)) as one function?? why?? and what does d/dt exactly mean?? (they get multiplied to a function that needs to be differentiated and I wanted to...

Hi, I just want to see if I understood this. Since the geometric product is associative and so on we can write for two multivectors A and B given by A= \alpha_{0}+\alpha_{1}e_{1}+\alpha_{2}e_{2}+\alpha_{3}e_{1}\wedge e_{2} B= \beta_{0}+ \beta_{1}e_{1}+\beta_{2}e_{2}+\beta_{3}e_{1}\wedge e_{2}...

1. Prove a) r=(u*v)=r*u+r*v and b) d/dt(r*s)=r*ds/st+dr/dt*s 2. Homework Equations : b) dr/dt=lim t->0=Δr/Δt and Δr=r(t+Δt)-r(t) 3. Attempt at the solution: Okay, so I was able to work out part a but I'm not quite sure how to start part b. Could anyone point me toward a useful resource to...

Homework Statement A and B are two unit vectors in the x-y plane. A = <cos(a), sin(a)> B = <cos(b), sin(b)> I need to derive the trig identity: sin(a-b) = sin(a) cos(b) - sin(b) cos (a) I'm told to do it using the properties of the cross product A x B Homework Equations A x B =...

Homework Statement Vectors A & B lie in an xy plane. A has a magnitude 7.4 and an angle 142(deg) with respect to the +x direction. B has components (-6.84i, -7.37j) B) What is the angle between the -y axis and the direction of the Cross product between A and B? Homework Equations Cross...

Hi there. I was following a deduction on continuum mechanics for the invariant nature of the first two laws of thermodynamics. The thing is that this deduction works with an identity, and there is something I'm missing to get it. I have the vector product: ##\vec \omega \times grad \theta##...

Let $0<a_i<\pi$, $i=1,\,\cdots,\,n$ and let $a=\dfrac{a_1+\cdots+a_n}{n}$. Prove that $\displaystyle \prod_{i=1}^{n} \left(\dfrac{\sin a_i}{a_i}\right)\le \left(\dfrac{\sin a}{a}\right)^n$.

Hey JO, I'm reading a book on geometric algebra and in the beginning (there was light, jk) a simple calculation is shown: Geometric product is defined as: ab = a \cdot b + a \wedge b or ba = a \cdot b - a\wedge b Now (a\wedge b)(a \wedge b)=(ab-a \cdot b)(a\cdot b - ba) =-ab^{2}a-(a...

Alright, so I was reading up on tensors and such with non-Cartesian coordinate systems all day but now I'm a bit tired an confused so you'll have to forgive me if it's a stupid question. So to express the dot product in some coordinate system, it's: g(\vec{A}\,,\vec{B})=A^aB^bg_{ab} And, if...

I am reading Chapter 2: Vector Spaces over $$\mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C}$$ of Anthony W. Knapp's book, Basic Algebra. I need some help with some issues regarding the general UMP-based definition of external and internal direct products ... ... On page 63, Knapp...

So, is there anyway to make the dot product change linearly? What I mean by this is when the angle is 45 degrees, I want it to be 0.5 instead of 0.7071 as you can see in this image: Instead I want 45 degrees to be 0.5, 60 degrees to be 0.33 and 30 degrees to be 0.66. Same would apply for...

Hello people. I'm thinking of using something like Twaron or Nomex fabric for a new application but one of the draw backs is the heat insulation property of these fabrics and I need to get around that and hopefully (in a wish upon a star kind of way) add a bit to the heat it can take. I was...

Hey guys, I was just doing some independent study on products of series and I'm trying to understand/derive the following form of the Cauchy product of series: \left(\sum_{n=0}^{N} a_{n}\right) \left(\sum_{m=0}^{N} b_{m}\right) = \sum_{n=0}^{N} \left(\sum_{k=0}^{n} a_{k}b_{n-k}\right)...

I'm reading through Douglas Gregory's Classical Mechanics, and at the start of chapter 6 he says that m \vec{v} \cdot \frac{d\vec{v}}{dt} = \frac{d}{dt}\left(\frac12 m \vec{v} \cdot \vec{v}\right), but I'm not sure how to get the right hand side from the left hand side. If someone could point...

Homework Statement Find x x = (1 + \frac{1+i}{2})(1 + (\frac{1+i}{2})^{2})(1 + (\frac{1+i}{2})^{2^{2}})...(1 + (\frac{1+i}{2})^{2^{n}}) Homework Equations Complex algebra equationsThe Attempt at a Solution Me again and another olympic question. I've tried some trigonometric substitutions...

I would like to convert: \frac{1}{1+x^{2m}} into a sum of terms. Preferably m terms but 2m terms would be OK. I start off with: \frac{1}{1+x^{2m}}=\frac{1}{\Pi^k_1(x-e^{i\frac{2k-1}{2m}\pi})(x-e^{-i\frac{2k-1}{2m}\pi})}=\frac{1}{\Pi^k_1 (x^2-2x \cdot cos \frac{2k-1}{2m}\pi +1)} where k = 1...

Hi all, I'm trying (and failing miserably) to understand tensors, and I have a quick question: is the inner product of a rank n tensor with another rank n tensor always a scalar? And also is the inner product of a rank n tensor with a rank n-1 tensor always a rank n-1 tensor that has been...

Hello, I'm working with a problem in linear elasticity, and I have to calculate the strain energy function as follows: 2W = σijεij Where σ and ε are symmetric rank 2 tensors. For cartesian coordinates it is really easy because the metric is just the identity matrix, hence: 2W = σxxεxx +...

Hi there. I wanted to demonstrate this identity which I found in a book of continuum mechanics: ##curl \left ( \vec u \times \vec v \right )=div \left ( \vec u \otimes \vec v - \vec v \otimes \vec u \right ) ## I've tried by writting both sides on components, but I don't get the same, I'm...

The uuu hadron doesn't violate Pauli's exclusion principle presumably because there is color. But even without color, can't the uuu exist if spatial wavefunctions are different? Suppose one u quark is located at r1, another at r2, and another at r3, and say that all three u quarks have spin up...

Homework Statement Suppose \vec{u}, \vec{v} and \vec{w} are vectors in an inner product space such that: inner product: \vec{u},\vec{v}= 2 inner product: \vec{v},\vec{w}= -6 inner product: \vec{u},\vec{w}= -3 norm(\vec{u}) = 1 norm(\vec{v}) = 2 norm(\vec{w}) = 7 Compute...

While studying Yang-Mills theory, I've come across the statement that there exists a positive-definite inner product on the lie algebra ##\mathfrak g## iff the group ##G## is compact and simple. Why is this true, and how it is proved?

With groups, one often seeks to create larger groups out of smaller groups, or the reverse: break down large groups into easier-to-understand pieces. One construction often employed in this regard is the direct product. The normal way this is done is like so: The direct product of two groups...

Hey guys I'm new here and I've been using MathType for all on-screen math. For some reason PF doesn't recognize the built-in Dirac's bra-ket notation. (i.e. <\psi|) So I've included my equations and solution in the format of images, hopefully it isn't a problem. Homework Statement Proof...

[SIZE="4"]Definition/Summary The product rule is a method for finding the derivative of a product of functions. [SIZE="4"]Equations (fg)'\ =\ f'g\ +\ fg' (fgh)'\ =\ f'gh\ +\ fg'h\ +\ fgh' [SIZE="4"]Extended explanation If a function F is the product of two other functions f and...

[SIZE="4"]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...

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)