Projection Definition and 434 Threads

Hi all: I am confused about why photon only has two projection along the z-direction. This confusion came from what I read in shankar "printciple of quantum mechanics". In the chapter of field quantization, he explain that because photon must satisfy transverse condition. wave function...

please can u do this sum for me...really urgent situation find the equation of the orthogonal projection of the line x+1/1 = 2y/-1 = z+1/2 on the plane x+2y+z=12 thanks in advance

A force F of 6 units acts in the direction 30 degrees west of north. An object is constrained to move north-westerly, that is, 45 degrees west of north. (a) Sketch the force vector roughly to scale on a set of axes that has the positive y axis pointing north, and write F using exact values...

I am trying to find the equation that will let me do an inverse azimuthal equidistant projection from the values contained within a 3-ball onto the surface of a 3-sphere. I have found plenty of information concerning 3-spheres, but nothing having to do with this kind of projection. I have also...

Stereographic Projection for "general" surfaces First off, sorry if this is in the wrong forum. I came across this while studying computer vision, but it's of a somewhat mathematical nature. Please move it if it's in the wrong place. In the book I'm reading*, stereographic projection is used...

Is a projection a quotient map? I think a quotient map is an onto map p:X-->Y (where X and Y are topological spaces) such that U is open/closed in Y iff (p)-1(U) is open/closed in X. And a projection is a map f:X-->X/~ defined by f(x)=[x] where [x] is the equivalent class (for a...

let pf(x)= sum( from i=1 to k) <x, ui>ui, show pf is a projection. Ive tried to show this fact myself but i failed. Please some one help me out. thanks Note ui = u1...un an orthogonal basis of V where V is a vector space.

Find the matrices of the transformations T which orthogonally project a point (x,y,z) on to the following subspaces of R^3. (a) The z-axis (b) the straight line x=y=2z (c) the plane x+y+z=0 (a) is easy just the matrix [0 0 0;0 0 0;0 0 1] as for (b) and (c) i have no idea how to work them out...

Why stereographic projection preserves angles between curves but does not preserve area?

Alright so I am trying to find the projection matrix for the subspace spanned by the vectors [1] and [2] [-1] [0] [1] [1] I actually have the solution to the problem, it is ... P = [ 5 1 2 ] (1/6) [1 5 -2]...

Homework Statement How would i go about solving Proju(Proju(v))=Proju(v) Just a note Proju(v) means the projection of v onto you Homework Equations The Attempt at a Solution how would i go about solving this is mathematical terms, it is obvious when you do it...

Homework Statement A firefighting crew uses a water cannon that shoots water at 25.0 m/s at a fixed angle of 53.0 degrees above the horizontal. The firefighters want to direct the water at a blaze that is 10.0 m above ground level. How far from the building should they position their...

Figures that the only problem I have trouble with is the one the book considers to be "easy": Homework Statement The speed of a projectile when it reaches its maximum height is one half its speed when it is at half its maximum height. What is the initial projection angle of the projectile...

Homework Statement After a package is ejected from the plane, how long will it take for it to reach sea level from the time it is ejected? Assume that the package, like the plane, has an initial velocity of 220 mph in the horizontal direction. If the package is to land right on the island...

Many books on QM state this so called von Naumann projection postulate i.e. that after the measurement system is in eigenstate of operator whose eigenvalue is measured. But in Landau Quantum Mechanics in chapter 7, author explicitly says that after the measurement system is in a state that...

hello again, I'm once again stumped, i was asked to find the rank and nullity of the projection u onto v so here is the given: T(u)=ProjvU, where v = <2,4> and this is what i did: let u = <u1 , u2> and plugged everything in the projection formula and ended up with < 4 + 2(u1) , -16 +...

Homework Statement Use the scalar projection to show that a distance from a point P(x1, y1) to the line ax + by + c = 0 is \frac{ax1 + by1 + c}{\sqrt{a^2 + b^2}}Homework Equations scalar projection = \frac{a . b}{|a|} The Attempt at a Solution The first thing that I did was to say that b =...

Homework Statement Hello! :smile: Find the equation of the projection of the line \frac{x}{4}=\frac{y-4}{3}=\frac{z+1}{-2} of the plane x-y+3z+8=0. So the line projects itself on the plane... Homework Equations The Attempt at a Solution First I find equation of line which...

Homework Statement The projection of the vector V onto (a,b) = (a,b) The projection of the vector V onto (-b,a) = (-b,a) Describe V in terms of a and b Homework Equations The Attempt at a Solution I let V=(x,y) then place that into the projection equation for each to get...

Homework Statement Find the projection of the intersection between the two surfaces S1: z = 4-x^2 - y^2 and S2: 4x^2y = 1 (x>0) in the xy-plane 2. The attempt at a solution 4-x^2 - y^2 = 4x^2y -1 Is this all I need to do?

Image a kite (1 m wide, 3 m high, both crossing at a third of the height). Also imagine a digital camera (800x600 pixel with a horizontal field of view of 45°). After launching the kite a photo is taken with the camera. How can I easily calculate the exact position *and* rotation of the...

[SOLVED] Quick Question: Is this matrix an orthogonal projection? Homework Statement P=[0 0 ] [11] Homework Equations The Attempt at a Solution Its orthogonal if the null space and range are perpendicular. Range=[0 ] [x+y] null space=[x

Hi all Not sure if I posted in the right spot but my question is in regards to optics and projection. To make a little less confusing I will try to explain my dilemma. I have a triplet lens out of an old crt projector where the rear focal point is about 10mm form the rear lens surface. So...

Supposing we have a vector space V and a subspace V_1\subset V. Suppose further that we have two different direct sum decompositions of the total space V=V_1\oplus V_2 and V_1\oplus V_2'. Given the linear projection operators P_1, P_2, P_1', P_2' onto these decompositions, we have that...

Let X be a norm space, and X=Y+Z so that Y\cap Z=\{0\}. Let P:X->Z be the projection y+z\mapsto z, when y\in Y and z\in Z. I see, that if P is continuous, then Y must be closed, because Y=P^{-1}(\{0\}). Is the converse true? If Y is closed, does it make the projection continuous? If...

Problem: Let \vec{x} and \vec{y} be vectors in Rn and define p = \frac{x^Ty}{y^Ty}y and z = x - p (a) Show that \vec{p}\bot\vec{z}. Thus \vec{p} is the vector projection of x onto y; that is \vec{x} = \vec{p} + \vec{z}, where \vec{p} and \vec{z} are orthogonal components of \vec{x}...

[SOLVED] Hilbert space &amp; orthogonal projection Homework Statement Let H be a real Hilbert space, C a closed convex non void subset of H, and a: H x H-->R be a continuous coercive bilinear form (i.e. (i) a is linear in both arguments (ii) There exists M \geq 0 such that |a(x,y)| \leq...

[SOLVED] Projection Theorem Homework Statement If M is a closed subspace of a Hilbert space H, let x be any element in H and y in M, then I have to show that \|x-y\| =\inf_{m\in M}\|x-m\| implies (equivalent to) that x-y\in M^{\perp} The Attempt at a Solution I have shown...

Are projections always continuous? If they are, is there simple way to prove it? If P:V->V is a projection, I can see that P(V) is a subspace, and restriction of P to this subspace is the identity, and it seems intuitively clear that vectors outside this subspace are always mapped to shorter...

[SOLVED] simple projectile motion.. Homework Statement A projectile is fired in such a way that its horizontal range is equal to 3 times its maximum height. What is the angle of projection? Homework Equations whole bunch for proj motion. The Attempt at a Solution I know that Dx =...

Projection of one vector on another?? Can anyone explain how to find the projection of one vector along another? I thought it was scalar (dot) product, but then I realized it WASN'T. What is this then? Anyone explain?

Homework Statement If at height of 40 m the direction of motion of a projectile makes an angle 45 degrees with the horizontal, then what is its initial velocity and angle of projection? Homework Equations The Attempt at a Solution

My teacher gave me this problem today and I have tried everything I know but I still haven't found the right answer. If anyone knows how to solve it, please share. Thanks At what projection angle will the range of a projectile equal its maximum height? Hint: 2 sin θ cos θ = sin 2 θ

Homework Statement How do I prove that if, |\vec{u_1}><\vec{u_1}| + |\vec{u_2}><\vec{u_2}| = I, where 'I' is the indentity matrix, that u_1 and u_2 are orthogonal and normalized? Can anybody get me started?

Golf problem... Ben is out at the practice range hitting golf balls. How much further will a golf ball with an initial speed of 75.0 m/s go when projected at 45.0 degree than when projected at 30.0 degree?

I noticed that it dosen't project to the origin of the plane from the north pole. However the projection describes it as mapping to the whole equitorial plane which is wrong! i.e take S^1 projecting to R. From the north pole the projection formula is y=-(x-a)/a however a can't be 0. So the...

Homework Statement 1.I have a vector defined by (v1,v2,v3). 2. I want to project this vector on a plane such that a point on that plane is defined by (p1,p2,p3).Also, the normal to the plane is given by (n1,n2,n3) 3.Can anyone help me to the projection of the vector on this plane...

The speed of a projectile when it reaches its maximum height is one half its speed when it is at half its maximum height. What is the initial projection angle of the projectile? Please help. Thanks.

Homework Statement My questions is this: How to find the orthogonal projection of vector y= (7,-4,-1,2) on null space N(A) Where A is a matrix A = \left(\begin{array}{cccc}2&1&1&3\\3&2&2&1\\1&2&2&-9\end{array}\right) Homework Equations A^TA\overline{x}=A^T\overline{y} The...

Urgent Stereographic projection question... Homework Statement Given the unit sphere S^2= \{x^2 + y^2 + (z-1)^2 = 1\} Where N is the Northpole = (0.0.2) the stereographic projection \pi: S^2 \sim \{N\} \rightarrow \mathbb{R}^2 carries a point p of the sphere minus the north pole N onto the...

Homework Statement A 6kg projectile is launced at an angle of 30 degrees to the horizontal and at initial spped of 40m/s. At the top of its flight, it explodes into 2 parts with masses 2 and 4 kg. The fragments move horizontally just after the explosion and the 2kg piece lands back at the...

I am studying for exam and something does not make sense anymore: looking at projection matrix, how come P=P2 where P2 = A(ATA)-1ATA(ATA)-1AT = A(ATA)-1AT = P but then they also say that cancelations (like distributing inverse operation and having AA-1 = I type things) are possible only if A...

Hi there, Look at the topic from my textbook "Stereographic Projection". Please inform me if I have understood it correctly :) Lets take specific example from my textbook S' = \{(x,y,z)|x^2 + y^2 + (z-1)^2 = 1\} is a sphere where N = (0,0,2) and P = (x,y,z) can be viewed as steographic...

Homework Statement I am trying to find the matrix M that projects a vector b into the left nullspace of A, aka the nullspace of A transpose. Homework Equations A = matrix A ^ T = A transpose A ^ -1 = inverse of A e = b - A x (hat) e = b-p I know that the matrix P that projects...

Homework Statement Let V=Mn(F) be the space of all nxn matrices over F; define TA=(1/2)(A+transpose(A)) for A in V. Verify that T is not only a linear operator on V, but is also a projection. Homework Equations A is a projection when A squared=A. The Attempt at a Solution I don't...

Homework Statement A stone of mass 2 kg is projected horizontally with a speed of 20 m/s from a cliff which is 15 m above the ground. Find the energy possessed by the stone just before touching the ground. A. 400J B. 500J C. 600J D. 700J Homework Equations The Attempt at a...

There is many projection (or measurement) postulates in quantum mechanics axioms: von Neumann measurement, Luders postulate... But does anybody know sth. about DIRAC POSTULATE? Thx

Hi all I have searched on google already but couldn't find any good tutorials. I am talking about isometric projection from area (3D) to coordinate system (2D). Here is the ''easy'' example of the little house in 3D view http://img.photobucket.com/albums/v309/Andreii/3d2d.jpg . I know I...

Question (5.1, #26 -> Bretscher, O.): Find the orthogonal projection of \left[\begin{array}{c} 49 \\ 49 \\ 49 \end{array}\right] onto the subspace of \mathbb{R}^3 spanned by \left[\begin{array}{c} 2 \\ 3 \\ 6 \end{array}\right] and \left[\begin{array}{c} 3 \\ -6 \\ 2 \end{array}\right]...

I'm wondering if the projection of the empty set has been defined? several books I've read seem to regard it in different ways