Orthogonal Definition and 560 Threads

Homework Statement Find Orthogonal Trajectories of ##\frac{x^2}{a}-\frac{y^2}{a-1}=1## Hint Substitute a new independent variable w ##x^2=w## and an new dependent variable z ##y^2=z## Homework EquationsThe Attempt at a Solution substituting ##x## and ##y## I get...

The complex exponential form of cosine cos(k omega t) = 1/2 * e^(i k omega t) + 1/2 * e^(-i k omega t) The trigonometric spectrum of cos(k omega t) is single amplitude of the cosine function at a single frequency of k on the real axis which is using the basis function of cosine, right? The...

Homework Statement x = <0, 10, 0> v1 = <4, 3, 0> v2 = <0, 0, 1> Project x onto plane spanned by v1 and v2 Homework Equations Projection equation The Attempt at a Solution I took the cross product k = v1xv2 = <3, -4, 0> I projected x onto v1xv2 [(x*k)/(k*k)]*k = <-4.8, 6.4, 0 = p I finished...

Homework Statement Given ##a\neq b## vectors of ##\mathbb{R}^n##. Determine ##c## which lies in the line segment ##[a,b]=\{a+t(b-a) ; t \in [0,1]\}##, such that ##c \perp (b-a)##. Conclude that for all ##x \in [a,b]##, with ##x\neq c## it is true that ##|c|<|x|##. Homework Equations The first...

I am exploring Gaussian integers in terms of roots, powers, primes, and composites. I understand that multiplying two integers with norm 5 result in an integer with norm 25. I get the impression that there are twelve unique integers with norm 25, and they come in two flavors: (1) Four of them...

A problem that I have to solve for my Linear Algebra course is the following We are supposed to use Mathematica. What I have done is that I first checked that A is symmetric, i.e. that ##A = A^T##. Which is obvious. Next I computed the eigenvalues for A. The characteristic polynomial is...

Given a real-valued matrix ## \bar{B}_2=\begin{bmatrix} \bar{B}_{21}\\ \bar{B}_{22} \end{bmatrix}\in{R^{p \times m}} ##, I am looking for an orthogonal transformation matrix i.e., ##T^{-1}=T^T\in{R^{p \times p}}## that satisfies: $$ \begin{bmatrix} T_{11}^T & T_{21}^T\\ T_{12}^T...

I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of Example 1.5.3 ... Duistermaat and Kolk"s Example 1.5.3 reads as follows:In the above example we read the...

Hi all. So to start I'll say I'm just dealing with functions of a real variable. In my linear algebra courses one thing was drilled into my head: "Algebraic invariants are geometric objects" So with that in mind, is there any geometric connection between two orthoganal functions on some...

Homework Statement Derive the relativistic Euler equation by contracting the conservation law $$\partial _\mu {T^{\mu \nu}} =0$$ with the projection tensor $${P^{\sigma}}_\nu = {\delta^{\sigma}}_\nu + U^{\sigma} U_{\nu}$$ for a perfect fluid. Homework Equations $$\partial _\mu {T^{\mu \nu}} =...

Homework Statement Being F = (1,1,-1), the orthogonal projection of (2,4,1) over the orthogonal subspace of F is: a) (1,2,3) b) (1/3, 7/3, 8/3) c) (1/3, 2/3, 8/3) d) (0,0,0) e) (1,1,1) The correct answer is B Homework Equations The Attempt at a Solution Using the orthogonal projection...

Hello, I Have a non-Hermitian Hamiltonian, which is defined as an ill-condition numbered complex matrix, with non-orthogonal elements and linearily independent vectors spanning an open subspace. However, when accurate initial conditions are given to the ODE of the Hamiltoanian, it appears to...

Hello, I want to show $ T^{*}(Pr_{C}(y)) = Pr_{T^{*}(C)}(T^{*}y)$ where $T \in B(H)$ and $TT^{*}=I$ , $H$ is Hilbert space and $C$ is a closed convex non empty set. but i don't know how to start, or what tricks needed to solve this type of problems. also i want know how to construct $T$ to...

Hi I can't see the error in this can someone please explain where I went wrong? A man is in a spaceship traveling at a constant velocity He makes 2 identical tubes of length L with a mirror at one end, tube a and tube b He has a single light bulb. Next to the bulb is a detector. He carefully...

Hi everyone, I am trying to self study some general relativity however I met some problem in the contravarient and covarient basis. In the lecture, or you can also find it on wiki page 'curvilinear coordinates', the lecturer introduced the tangential vector ei =∂r/∂xi and the gradient vector ei...

Hiya again, I am trying to solve this problem, I thought I got somewhere, but kinda stuck. The graph of y^2=x^3 is called a semicubical parabola. Determine the constant b so that the line y = -(1/3)x+b meets this graph orthogonally. I found the derivative of the curve by using implicit...

Consider a force-free particle moving on a geodesic with four-velocity v^\nu. The formula for the four-acceleration in any coordinate system is \frac{dx^\mu}{d\tau} = - \Gamma^\mu_{\nu\lambda} v^\nu v^\lambda Since the four-acceleration on the left side is orthogonal to the four-velocity, this...

Homework Statement I have recently come across the notation <ψ|Φ> in my notes and am not quite sure what it means. Some articles I have read online state that this is analogous to the dot product, except that this is the "dot-product" of 2 wave-functions. Would I then be right in saying that...

are spacelike and timelike orthogonal?what is the mathematical proof

I've been trying to get my head around Polarisation and how it achieves orthogonality. I'm not sure if this should be in Physics or Electrical Engineering Section. (Mods can move this where appropriate) I know that 2 EM wave with linear polarisations where one wave is shifted by π, they would...

An orthogonal basis set spanning R4 has four vectors, v1, v2, v3 and v4. If v1 and v2 are [ −1 2 3 0 ] and [−1 1 −1 0 ] find v3 and v4. Please explain this in a very simple way.

Homework Statement I have a linear transformation ##\mathbb{R}^3 \rightarrow \mathbb{R}^3##. The part that asks for a basis of eigenvectors I've already solved it. The possible eigenvectors are ##(1,-3,0), (1,0,3), (\frac{1}{2}, \frac{1}{2},1) ##. Now the exercise wants me to show that there is...

Homework Statement Show that the set GL(n, R) of invertible matrices forms a group under matrix multiplication. Show the same for the orthogonal group O(n, R) and the special orthogonal group SO(n, R). Homework EquationsThe Attempt at a Solution So I know the properties that define a group are...

Homework Statement Show that every matrix A ∈ O(2, R) is of the form R(α) = cos α − sin α sin α cos α (this is the 2d rotation matrix -- I can't make it in matrix format) or JR(α). Interpret the maps x → R(α)x and x → JR(α)x for x ∈ R 2 Homework EquationsThe Attempt at a Solution So I know...

Homework Statement Is it true in three dimensions that any two vectors perpendicular to a third one are parallel to each other? Homework Equations Dot product. The Attempt at a Solution I've come up with two vectors that were orthogonal to a third and found the angle between them using the...

$\tiny{s6.194.4.12.4.29}$ $\textsf{a. Find a nonzero vector orthogonal to plane through the points: }$ $\textsf{b. Find the area of the triangle PQR}$ \begin{align} \displaystyle &P(1,0,0)& &Q(0,2,0)& &R(0,0,3)\\ %&=\color{red}{\frac{1209}{28} } \end{align} $\textit{do what first?}$

Homework Statement find the vector in R3 that is a unit vector that is normal to the plane with the general equation x − y + √2z=5 [/B]Homework EquationsThe Attempt at a Solution so the orthogonal vector, I just took the coefficients of the general equation, giving (1, -1, √2)[/B] then...

I know that to find the projection of an element in R^n on a subspace W, we need to have an orthogonal basis in W, and then applying the formula formula for projections. However, I don;t understand why we must have an orthogonal basis in W in order to calculate the projection of another vector...

Homework Statement If a 3 x 3 matrix A is diagonalizable with eigenvalues -1, and +1, then it is an orthogonal matrix. Homework EquationsThe Attempt at a Solution I feel like this question is false, since the true statement is that if a matrix A is orthogonal, then it has a determinant of +1...

Homework Statement I have to find for which values of the real parameter ##b## the following plane is orthogonal to the following line: ##\pi : 5x + (2b - 1)y - (1 + 8b)z + 3 = 0## ##s : \begin{cases} x + z - 4 = 0 \\ x - 3y + z + 2 \end{cases}## Homework EquationsThe Attempt at a Solution For...

Homework Statement So, I'm doing a long exercise, you can check here the first part: https://www.physicsforums.com/threads/checking-if-the-following-lines-are-coplanar.885948/ The second part asks me to find, if one of the couple of lines are skew, the orthogonal line to two skew lines...

Homework Statement Show that the matrix ##P = \big{[} p_{ij} \big{]}## is orthogonal. Homework Equations ##P \vec{v} = \vec{v}'## where each vector is in ##\mathbb{R}^3## and ##P## is a ##3 \times 3## matrix. SO I guess ##P## is a transformation matrix taking ##\vec{v}## to ##\vec{v}'##. I...

I'd like to expand a 3D scalar function I'm working with, ##f(r,\theta,\phi)##, in an orthogonal spherical 3D basis set. For the angular component I intend to use spherical harmonics, but what should I do for the radial direction? Close to zero, ##f(r)\propto r##, and above a fuzzy threshold...

Could you view a discrete number, for instance a binary number, as a sort of orthogonal basis, where each digit position represents a new dimension? I see similarities between a binary number and for instance Fourier Transform, with each digit being a discrete function.

Hello, friends! Let us define the external measure of the set ##A\subset \mathbb{R}^n## as $$\mu^{\ast}(A):=\inf_{A\subset \bigcup_k P_k}\sum_k m(P_k)$$where the infimum is extended to all the possible covers of ##A## by finite or countable families of ##n##-paralleliped ##P_k=\prod_{i=1}^n...

Hi in my project their are three variables like speed ,depth of cut and feed and each variable have 5 values then please tell me which Orthogonal array is suitable for my design of experiments by using taguchi method And tell me detailed Thank you

Homework Statement Vector u, v, and x are not zero. Vector u + v will be perpendicular (orthogonal) to u-x if A. |u+v| = |u-v| B. |v| = |x| C. u ⋅ u = v ⋅ v, v = -x D. u ⋅ u = v ⋅ v, v = x E. u ⋅ u = v ⋅ v Homework Equations u⋅v = |u||v| cos θ The Attempt at a Solution [/B] Two vectors are...

Homework Statement I'm told to find the matrix Q of the matrix A Homework EquationsThe Attempt at a Solution So my problem is that in the answer key they have S = (1/3)... and I have no idea where this 1/3 comes from. I get an equivalent answer for X_1, X_2, and X_3 S = [X_1, X_2, X_3] but...

Could someone explain to me in simplest of terms what scale factor is when dealing with orthogonal vectors.

[Mentor's Note: Thread moved from homework forums] Where can I start to research this question? I did not take any course on Group theory before and I know almost nothing about the relationship with this pure maths and physics. I've decided to start with Arfken's book but I'm not sure. 1...

Homework Statement Let V be the 2-dim subspace of R^3 spanned by V1 = (1,1,1) and V2 = (-2,0,1). Find the orthogonal compliment Vperpendicular. 2. Relevant equation X + Y + Z = 0 -5X + Y + 4Z = 0 The Attempt at a Solution Firstly, I orthogonalize the basis for V and get the vectors (1,1,1)...

Trying to figure out the orthogonal trajectory of x^2 + y^2=cx^3 Here's what I got... but it does not match the books answer. I don't know where I am going wrong. I think I was able to differentiate the equation correctly in order to get the inverted reciprocal slope and then I may have flubbed...

Are the Eigenspectra (a spectrum of eigenvalues) and the Empirical Orthogonal Functions (EOFs) the same? I have known that both can be calculated through the Singular Value Decomposition (SVD) method. Thank you in advance.

We know that modes of vibration of an Euler-Bernoulli beam are given by eigenfunctions, with the natural frequency of each mode being given by its eigenvalue. Thus these modes are all mutually orthogonal.Can anything be said of the derivatives of these eigenfunctions? For example, I have the...

Homework Statement Show that the set {sin(nx)} from n=1 to n=∞ is orthogonal bases for L^2(0, π). Homework EquationsThe Attempt at a Solution Proof: Let f(x)= sin(nx), consider scalar product in L^2(0, π) (ƒ_n , ƒ_m) = \int_{0}^π ƒ_n (x) ƒ_m (x) \, dx = \int_{0}^π sin(nx)sin(mx) \, dx =...

Is the basis vector ##(i,0,1)## in the space ##V=##Span##((i,0,1))## with a standard inner product,over ##\mathbb{C}^3## orthogonal to itself? ##<(i,0,1),(i,0,1)> = i \cdot i + 0 \cdot 0 + 1 \cdot 1 = -1 + 1 = 0 ## The inner product (namely dot product) of this vector with itself is equal to...

Can an orthogonal matrix involve complex/imaginary values?

1. Okay, so I am going to prove that \int H_a\cdot H_bdv=0 Hint: Use vector Identities H is the Magnetic Field and v is the volume. Homework Equations this this[/B] k_bH_b=\nabla \times E_b k_aH_a=\nabla \times E_a k is the wave vector and E is the electric field The Attempt at a...

Homework Statement Hey, the no-cloning theorem states, that arbitrary quantum states cannot be cloned by any circuit. It is, however, possible to clone orthogonal states. What would a circuit performing this action look like? Homework Equations Relevant equations: I am assuming you all now...

Find the general form of an orthogonal 2 x 2 matrix = $ \begin{bmatrix}a&b\\c&d\end{bmatrix}$ I used $det(A)=\pm 1$ (from an earlier exercise) - and the special form of $ A_{2 \times 2}^{-1} = \frac{ \begin{bmatrix}d&-b\\-c&a\end{bmatrix}}{|A|} $ using $|A| = +1$ first, to get an...