Product Definition and 1000 Threads

Homework Statement Prove that if u and v are nonzero vectors, and theta is the angle between them then u dot product v = ||u|| ||v|| cos (theta). Consider the triangle with sides u ,v , and u-v. The Law of Cosines implies that ||u-v||^2 = ||u||^2 + ||v||^2 - 2||u|| ||v|| cos(theta). On the...

I got asked how the scalar product can be used to find the length of a vector? Could someone please explain

f_{k} \overset{m}{\rightarrow} f and g_{k} \overset{m}{\rightarrow} g over E. Then: a) f_{k} + g_{k} \overset{m}{\rightarrow} f+g over E b) If | E | < + \infty, then f_{k} g_{k} \overset{m}{\rightarrow} fg over E. Show that the hiphotesis | E | < + \infty is neccesary c) Let \{...

Find the product of 3 distinct real numbers $a, b, c$ if they satisfy the system of equations $a^3=3b^2+3c^2-25$ $b^3=3c^2+3a^2-25$ $c^3=3a^2+3b^2-25$

Homework Statement See attached image Homework Equations The Attempt at a Solution For the first half of the question, ordered pairs would be (1, [1]), since 1 and [1] are the multiplicative identities in these rings. but no matter how many times we add (1, [1]) to itself...

Correct me if I'm wrong here but it is my understanding that vector spaces are given structure such as inner products, because it allows us to use these structured vector spaces to describe and analyse physical things with them. So physical properties such as 'distance' cannot be analysed in...

Let A be a Hermitian operator with n eigenkets: A|u_i\rangle = a_i |u_i\rangle for i=1,2,...,n. Suppose B is an operator that commutes with A. How could I show that \langle u_i | B | u_j \rangle = 0 \qquad (a_i \neq a_j)? I have tried the following but not sure how to proceed: AB -...

Let $p$ be the sum and $q$ be the product of all real roots of the equation $x^4-x^3-1=0$. Prove that $q<-\dfrac{11}{10}$ and $p>\dfrac{6}{11}$.

Warning This product attracts every other piece of matter in the Universe, including products of other manufacturers, with a force proporitional to the product of the masses and inversely proportional to the square of the distance between them.Handle with Extreme Care This product contains...

Homework Statement I need to answer a bunch of topological questions based on the cartesian product of two sets, but I'm not entirely sure how to graph them out. I have A = [1,2)U{3} and B = {1, (1/2), (1/3), ...}U[-2,-1). S = A x B, and I need the graph of S. Could anyone help me with...

Consider the following commutator for the product of the creation/annihilation operators; [A*,A] = (2m(h/2∏)ω)^1 [mωx - ip, mωx + ip] = (2m(h/2∏)ω)^1 {m^2ω^2 [x,x] + imω ([x,p] - [p,x]) + [p,p]} Since we have the identity; [x,p] = -[p,x] can one assume that.. [x,p] - [p,x] =...

Need someone to check my answer please. Consider a 4 input, 1 output digital system (W,X,Y,Z, and f respectively) . Design a POS circuit with any number of inputs such that f(W,X,Y,Z) = M(0,2,4,9,13) + D(6,14). First fill in the Truth Table, then find the minimum product of sums equation using...

can you show me derive cross product from dot product?

hey all well the title says it all. if i want to take the cross product of two matrices, how do i do it? any help, advice, etc. is very appreciated! thanks

How do you take the cross product of two 4-Vectors? \vec{r} = \left( \begin{array}{ccc}c*t & x & y & z \end{array} \right) \vec{v} = \left( \begin{array}{ccc}c & vx & vy & vz \end{array} \right) \vec{v} \times \vec{r} = ?

Homework Statement Use (the bac-cab rule) to expand this triple product:L = mr x (ω x r) If r is perpendicular to ω, show that you obtain the elementary formula, angular momentum = mvr. (The bold letters are vectors.) Homework Equations A X (B X C) = (A\cdotC)B - (A\cdotB)C...

Let A be a proper subset of X, and let B be a proper subset of Y . If X and Y are connected, show that X × Y − A × B is connected. Attempt: Proven before in my book, I know that since X and Y are each connected that X x Y is also connected. Keeping that fact in mind. Pf: Assume (X x Y) -...

Homework Statement Compound A mass - 11.0g solubility in ethanol at 78 deg C - 0.53 g/mL solubility in ethanol at 0 deg C - 0.08 g/mL Impurity B mass - 2.5 g solubility in ethanol at 78 deg C - 0.82 g/mL solubility in ethanol at 0 deg C - 0.12 g/mL Impurity C mass - 2.5 g solubility in...

Show that $3^{2008}+4^{2009}$ can be written as product of two positive integers each of which is larger than $2009^{182}$

Homework Statement Hi all, Here's the problem: Prove, in tensor notation, that the triple scalar product of (A x B), (B x C), and (C x A), is equal to the square of the triple scalar product of A, B, and C. Homework Equations The Attempt at a Solution I started by looking at the triple...

Show that TS^1 is diffeomorphic to TM×TN. (TS^1 is the tangent bundle of the 1-sphere.) We can use the theorem stating the following. If M is a smooth n-manifold with or without boundary, and M can be covered by a single smooth chart, then TM is diffeomorphic to M×ℝ^n. Clearly, I must be...

Write out the minimal Product of Sums (POS) equation with the following Karnaugh Map. Just need someone to check my work please. I am questioning my self on my grouping. Did I group correctly or should I have grouped the bottom left 0 and D versus the 0 in the group of 8? Thanks for your time...

okay so I know that the area of the triangle is half the area of the parallelogram, ill try using pictures because this is a bit confusing to describe only with words: for example we have this http://farside.ph.utexas.edu/teaching/301/lectures/img243.png and then if we use the cross product of...

Awesome thanks.. Mind checking this as well? Minimize Sum of Products equation given the following K map. My Answer: \bar{y} \bar{w} + wx + y\bar{z}w + yw\bar{x}

Just need someone to check my work. Couldn't find the problem via Google. $f$(W,X,Y,Z) M (0,1,2,7,12,15) + d(3,13). 1)Find the minimum Product of Sums equation using a K-Map. 2)Draw a schematic of a minimized circuit implementing the logic using NOR gates. 1) My Answer: (\bar{w} +...

I'm following the book ''introduction to electrodynamics by D.J. Griffiths''. As he has written that the formula ''integral of product of infintesimal volume with the square of electric field'' gives us the energy contained in a charge configuration that is always positive because we're...

i have a vector xK where k is the unit vector perpendicular to other unit vectors i and j when i multiply a force which has 5k for instance another which has ( 3 i + 4 j ) i multiply 5k by 3i then 5k by 4j right ? the answer would be ( 15 j - 20 i ) right ?

Sorry guys, I have some differential topology homework, and I may be asking a lot of questions in the next few days. Problem Statement Suppose M_1,...,M_k are smooth manifolds and N is a smooth manifold with boundary. Then M_1×..×M_k×N is a smooth manifold with a boundary. Attempt Since...

Homework Statement My textbook says: The length of the cross product a x b is equal to the area of the parallelogram determined by a and b. How can a length equal and area? They have different units?

So in a given amount of solution, the equilibrium constant is the product of the concentrations of the ionic species present, raised to the power of the stoichiometric coefficient. Wouldn't it make more sense if it was the sum of the concentrations though,since basically the net number of ions...

Hi, I am trying find the simplified expression of this: ∇(E \cdot E) Where E is the electric field that can written as E_{0}(exp(i(kx-ωt)) I know that since the two vectors are the same => E \cdot E = ||E||^{2} Do I take the gradient of the magnitude then? It just doesn't feel...

Homework Statement if v x w = <5,5,-2> (v cross w) and v * w = 6 (v dot w) then what is the tan(θ) between the two vectors v and w? The Attempt at a Solution well I was thinking v x w = |v||w|sinθ as well as v dot w (v*w) = |v|w|cosθ divide one equation by the other...

Homework Statement A constant force of 1i - 5j -8k moves (1,-4,2) (-3,2,-1), what is the work done on the particle? Homework Equations Avector*Bvector=ABsinθ ?? I think? The Attempt at a Solution I really am quite lost... but I found the coordinates for the position vector...

The problem is: Let A be a real m x n matrix and let x be in R^n and y be in R^m (n and m dimensional real vector spaces, respectively). Show that the dot product of Ax with y equals the dot product of x with A^Ty (A^T is the transpose of A). The way I went about starting this problem is to...

I am new to tensor notation, but have known how to work with vector calculus for a while now. I understand for the most part how the Levi-Civita and Kronecker Delta symbol work with Einstein summation convention. However there are a few things I'm iffy about. For example, I have a problem where...

Hello, I am confused how vectors that are coplanar will give a triple product of zero? Or is it the case that all 3 vectors must be coplanar for a triple product of zero, or is 2 sufficient? I.e. the vector being dotted with one of the vectors being crossed in the same plane, will this...

Hi MHB, It's me again...I find this problem to be very interesting yet very difficult to me. I tried to approach it using the Vieta's formula, knowing the given function $p(x)$ has only one real root and 4 complex roots, where I let the 4 complex roots be $a\pm bi$ and $c\pm di$, but it failed...

Just like the title says, would that technically be true? I know the cross product is normal to the plane of the two vectors being crossed, which would make it z. However, since the angle between two vectors is 0, sin (0) = 0...

Homework Statement Assume that you are given differentiable function f(t) and g(t). Find a formula for the derivative of the cross product u(f(t)) x v(g(t)). Homework Equations d/dt(u(t) x v(t)) = (u'(t) x v(t) + u(t) x v'(t) The Attempt at a Solution So in this case I was thinking...

use this one "∏" anyone knows how to use it?

In QM the tensor product of two independent electron's spin state vectors represents the product state which represents the possible unentangled states of the pair. I don't understand why the tensor product produces that result. |A⟩=|a⟩⊗|b⟩

Homework Statement If the product of the numbers R and 11/S is the same as their sum, find the value of S. Homework Equations N/A The Attempt at a Solution I am suspecting that the only set of 2 numbers that have the same sum and product is 2 and 2. So I guess R is 2, 11/S is...

So, I have an equivalence I need to prove, but I think I'm having trouble understanding the problem at a basic level. The problem is to prove that the inner product of a and b equals 1/4[|a+b|^2 - |a-b|^2] (a, b in C^n or an n-dimensional vector space with complex elements). I don't...

$$ \frac{(\dot{\mathbf{r}}\times\ddot{\mathbf{r}}) \times\dot{\mathbf{r}}}{\lvert\dot{\mathbf{r}} \rvert\lvert\dot{\mathbf{r}}\times\ddot{\mathbf{r}}\rvert} $$ How do I take that dot product of the expression of above with itself?

Let V be a real vector space. Suppose to each pair of vectors u,v ε v there is assigned a real number, denoted by <u, v>. This function is called a inner product on V if it satisfies some axioms. 1. What does refers by "this function"? Is it "<u, v>"? If it is then How we can call it's a...

Hi, The following equations are from linear regression model notes but there is an aspect of the matrix algebra I do not get. I have, \mathbf{y} and \tilde{\beta} are a mx1 vectors, and \mathbf{X} is a mxn matrix. I understand the equation...

Homework Statement Calculate the net torque about O at P,assuming that a 30-kg mass is attached at P [Figure 21(B)]. The force Fg due to gravity on a mass m has magnitude 9.8m m/s2 in the downward direction. Homework Equations The torque about the origin O due to a force F act- ing on an...

I am reviewing some material on Laplace Transforms, specifically in the context of solving PDEs, and have a question. Suppose I have an Inverse Laplace Transform of the form u(s,t)=e^((as^2+bs)t) where a,b<0. How can I invert this with respect to s, giving a function u(x,t)? Would the inverse...

Homework Statement Calculate the cross product of (3u+4w)xw assuming that uxv=<1,1,0>, uxw=<0,3,1), vxw<2,-1,-1)Homework Equations Possible Relevant eqation: i) wxv=-vxw ii)vxv=0 iii)vxw=0 if and only w= λv for scalar λ or v=0 iV)(λv)xw=vx(λw)=λ(vxw) V) (u+v)xw= uxw+vxw ux(u+w)=uxv+uxw The...

Hi I thought about a question , say we have a sphere into which D D or D T fusion takes place , now that forms the end product which differs according to the reactants used but the thing that interests me here is that the fusion end product is always heavier than each of the products that made...