Product Definition and 1000 Threads

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...

Hello MHB. I need help with the follwoing: Evaluate $\displaystyle P=\prod_{j=1}^{1004}\cos\left(\frac{2\pi j}{2009}\right)$. I think it would be easier to first evaluate $\displaystyle Q=\prod_{j=1}^{2008}\cos\left(\frac{2\pi j}{2009}\right)$. Knowing the value of $Q$ we can easily find the...

momentum, position vector dot (scalar) product "action" Hello, I was playing with single mass point classical mechanics, when I realized that the dot product of the position vector and momentum vector, p.r , has action dimension. Furthermore, its time derivative, d/dt(p.r) = F.r + p.v, has...

Integrating a product of two functions - one "lags" the other I am wondering if there is a way to integrate the following function without first expanding the brackets: \int\limits_{x1}^{x2} x^2\left(x-a\right)^2\,dx The idea behind the question is a bit more complex than I am letting on...

Why does the cross product produce a vector and why is that vector perpendicular to the other vectors? I understand how to calculate a cross product, but why for instance is the cross products of two vectors another vector that is perpendicular to it. Can you prove or explain this to me in...

Homework Statement A=(x,3,1) ,B=(x,-x,2) Determine the value of x if the vector perpendicular to A and B is given by C=(10,-4,-4) Homework Equations The Attempt at a Solution Find A cross B , let A cross B be D . Then D cross C = zero (since they are perpendicular to both A and...

Hello, It may be trivial to many of you, but I am struggling with the following integral involving two spheres i and j separated by a distance mod |rij| ∫ ui (ρ).[Tj (ρ+rij) . nj] d2ρ The integration is over sphere j. ui is a vector (actually velocity of the fluid around i th...

Basically I don't know anyone in real life that can help me with this, so I need help checking to see if my answers are correct :) PART A 11) Find the vertex of f(x) = -2x^2 - 8x + 3 algebraically. My Answer: (-2,0) 12) Multiply and simplify: (6 - 5i) (4 + 3i) My Answer: 39 - 2i

Homework Statement Does the Cauchy Schwarz inequality hold if we define the dot product of two vectors A,B \in V_n by \sum_{k=1}^n |a_ib_i| ? If so, prove it. Homework Equations The Cauchy-Schwarz inequality: (A\cdot B)^2 \leq (A\cdot A)(B\cdot B) . Equality holds iff one of the vectors...

Is this correct in infinite dimensional Hilbert spaces? ## (\hat{A}_1 \otimes \hat{A}_2)^{-1}=\hat{A}^{-1}_1 \otimes \hat{A}^{-1}_2 ## ## (\hat{A}_1 \otimes \hat{A}_2)^{\dagger}=\hat{A}^{\dagger}_1 \otimes \hat{A}^{\dagger}_2 ## ## (\hat{A}_1 +\hat{A}_2) \otimes \hat{A}_3=(\hat{A}_1 \otimes...

a and b are two vectors and x is the angle between them. ||axb|| = ||a||||b||sinx ------(1) ||axb|| = ||a||||b||||sinx|| ------(2) which one is correct? why?

Homework Statement There are two locations for the lens along the optical bench that will focus an image on the screen. Find one of these locations. Once you have the image in sharp focus, take measurements for: the object distance, do, the image distance, di, and the height of the image, hi...

1) Show that for $n >1$, $\displaystyle \prod_{k=1}^{\infty} \left( 1- \frac{z^{n}}{k^{n}} \right) = \prod_{k=0}^{n-1} \frac{1}{\Gamma\left[ 1-\exp (2 \pi i k/n) z\right]}$.2) Use the above formula to show that $ \displaystyle \prod_{k=1}^{\infty} \left(1- \frac{z^{2}}{k^{2}} \right) =...

Homework Statement Let V be the real vector space of all real symmetric n × n matrices and define the scalar product of two matrices A, B by (Tr (A) denotes the trace of A) Show that this indeed fulfils the requirements on a scalar product. Homework Equations Conditions for a scalar...

Homework Statement Consider the vector space of continuous, complex-valued functions on the interval [−∏, ∏]. Show that defines a scalar product on this space. Are the following functions orthogonal with respect to this scalar product? Homework Equations The Attempt at a...

I have always wondered: Is the product rule and addition rule for that matter axioms of the probability theory or can they actually be proven from more general principles? The reason I ask is, and it might be a bit silly, that I have always thought I missed out on something in probability...

Consider 2 atomic orbitals with wave function a: σ(r), b: μ(r) in a diatomic molecules. σ(r) (or μ(r)) is localized around an atom a (or b) and is relevant for the discussion of the molecular orbital. These orbitals are orthogonal and normalised. The creation operators are x, y and vacuum, |0>...

a =(x,y), b =(h,k) a cross b =? I have idea what to type on google. Is that doing like matrices , a cross b = xk-hy? thanks.

Homework Statement Write each polynomial as the product of it's greatest common monomial factor and a polynomial Homework Equations 8x^2+12x 6a^4-3a^3+9a^2 The Attempt at a Solution 4x(2x+3) a^2(6a^2-3a+9) I really don't understand what it's asking me for, how can I factor this...

Homework Statement L(θ) = ∏(θ/(2√xi)*e^(-θ√xi)),i=1, n Homework Equations The Attempt at a Solution -> θ2∏(1/(2√xi)*e^(-θ√xi)) taking natural log of both sides lnL(θ) = nlnθ + ln∏(1/(2√xi)*e^(-θ√xi)) = nlnθ + Ʃln(1/(2√xi)*e^(-θ√xi)) Ok so from what I understand the...

Homework Statement Let A be a vector perpendicular to every vector X. Show that A = O Edit: it is O not 0. (OH not zero) haHomework Equations So, we know if A and X are perpendicular then A(dot)X = 0 I see no reason why A would have to be equal to 0. Could X (not equal) 0? Could it be...

Amazon.com (USA) is quietly deleting all customer contributed product photos from its website. The deletion is scheduled to be complete by Aug. 31. If you browse the site you'll notice the option to upload customer images is gone from most of the pages. (Meanwhile, Amazon has announced that...

Hi, I was reading vector mechanics by beer and jhonston. I came across the equation wherein the cross prodcut of two vectors P and Q is given. It says P x Q = P x Q` . I am not bale to understand how dis is possible. Because as the vector Q changes even the angle teetha will change then how...

Understanding the use of Pythagorean theorem for the length and square of a vector and how this is the dot product of a vector with itself is no problem. I'm trying to look inside the meaning of the dot product of two different vectors and understand it. I can also accept (just following the...

I would like to take divergence of the following expression ∇.((xi × xj × xk)/r^3), which is a triadic. here × denote a dyadic product and r=mod(r vector) and xi, xj and xk are the components of r vector. So, the above eq. can also be written as ∇.((xixjxk ei×ej×ek)/r^3), where ei...

so, the magnitude of the cross product represents the area or volume enclosed by any 2 or 3 vectors respectively, but what does the direction represent? i get that the general direction is one that is perpendicular to all the vectors, but what does the actual direction represent? ie, why the...

I would like to know if the Jordan Product carries along with it the same properties as that of the cross product (i.e. associativity, commutable, left/right distributive)? If you've taken LA, I'm sure you know that professors require us to complete a project and I've chosen the Jordan Product...

Is there atopological space X such that XxX (the product space) is homeomorphic to the unit circle in the plane

I have two real symmetric matrices A and B with the following additional properties. I would like to know how the eigenvalues of the product AB, is related to those of A and B? In particular what is \mathrm{trace}(AB)? A contains only 0s on its diagonal. Off diagonal terms are either 0 or...