What is Vector: Definition and 1000 Discussions

Homework Statement Homework Equations v = I + j + k v = d/t The Attempt at a Solution I thought the answer was as simple as: v = 63i + 0j + 0k, since the car only has motion in one direction... ...but I got it wrong, so clearly I'm missing something here.

I use the ##(-,+,+,+)## signature. In the Schwarzschild solution $$ds^2=-\left(1-\frac{2m}{r}\right)dt^2+\left(1-\frac{2m}{r}\right)^{-1}dr^2+r^2d\Omega^2$$ with coordinates $$(t,r,\theta,\phi)$$ the timelike Killing vector $$K^a=\delta^a_0=\partial_0=(1,0,0,0)$$ has a norm squared of...

Homework Statement Calculate |u+v+w|, knowing that u, v, and w are vectors in space such that |u|=√2, |v|=√3, u is perpendicular to v, w=u×v. Homework Equations |w|=|u×v|=|u|*|v|*sinΘ The Attempt at a Solution [/B] Θ=90° |w|=(√2)*(√3)*sin(90°)=√(6) Then I tried to use u={√2,0,0}...

Homework Statement The position vector of a particle changes: Only by its module. Only by its direction. What can be said about the trayectory of the movement of the particle? Obtain the answer analitically. Homework Equations None. The Attempt at a Solution I think that the trayectory...

this is what is given so by addition $$\begin{bmatrix}x_1\\y_1\\5z_1\end{bmatrix} \oplus \begin{bmatrix} x_2\\y_2\\5z_2 \end{bmatrix} = \begin{bmatrix} x_1+x_2\\y_1+y_2\\5z_1+5z_2 \end{bmatrix} = \begin{bmatrix} X\\Y\\10Z \end{bmatrix}$$ uhmmmm really?

On the set of vectors $\begin{bmatrix} x_1 \\ y_1 \end{bmatrix}\in \Bbb{R}^2 $ with $x_1 \in \Bbb{R}$, and $y_1$ in $\Bbb{R}^{+}$ (meaning $y_1 >0$) define an addition by $$\begin{bmatrix} x_1 \\ y_1 \end{bmatrix} \oplus \begin{bmatrix} x_2 \\ y_2 \end{bmatrix} = \begin{bmatrix} x_1 + x_2 \\...

Homework Statement a) A point charge + q is placed at the origin. By explicitly calculating the relevant line integral, determine how much external work must be done to bring another point charge + q from infinity to the point r2= aŷ ? Consider the difference between external work and work...

Homework Statement Show that the only subspaces of ##V = R^2## are the zero subspace, ##R^2## itself, and the lines through the origin. (Hint: Show that if W is a subspace of ##R^2## that contains two nonzero vectors lying along different lines through the origin, then W must be all of...

So say our inner product is defined as ##\int_a^b f^*(x)g(x) dx##, which is pretty standard. For some operator ##\hat A##, do we then have ## \langle \hat A ψ | \hat A ψ \rangle = \langle ψ | \hat A ^* \hat A | ψ \rangle = \int_a^b ψ^*(x) \hat A ^* \hat A ψ(x) dx##? This seems counter-intuitive...

Good day everybody, I'm currently working on the Grover algorithm. You can also illustrate this process geometrically and that's exactly what I have a question for. In my literary literature one obtains a uniform superposition by applying the Hadamard transformation to N-qubits. So far that's...

Homework Statement You wish to row straight across a 63 meter-wide river. You can row at a steady 1.3 m/s relative to the water and the river flows at 0.57 m/s. In what direction should you head, and how long would it take you to cross the river? Homework EquationsThe Attempt at a Solution...

Homework Statement A 0.54 kg block of ice is sliding by you on a very slippery floor at 2.1 m/s. As it goes by, you give it a kick perpendicular to its path. Your foot is in contact with the ice block for 0.0034 seconds. The block eventually slides at an angle of 21 degrees from its original...

Please see the attached page to see what I'm talking about. In the top right paragraph, it states to use "the positive direction of the x-axis". It is given that ##θ_2=30°## and it shown visually at the bottom of the page. In the problem it's using -60° and I'm not sure how they're getting that.

Dear all, I am trying to find if these two sets are vector subspaces of R^3. \[V=\left \{ (x,y,z)\in R^{3}|(x-y)^{2}+z^{2}=0 \right \}\] \[W=\left \{ (x,y,z)\in R^{3}|(x+1)^{2}=x^{2}+1 \right \}\] In both cases the zero vector is in the set, therefore I just need to prove closure to addition...

In 2-D Cartesian coordinate system let's there exist a scaler field Φ(x1,x2) ,now we want to find how Φ changes with a curve which is described by the parameter(arc length) s dΦ/ds=(∂Φ/∂xi)dxi/ds Can we say for Cartesian coordinate system that along the curve at any s dxi always points in the...

Hi, I'm just starting to read Wald and I find the notion of the commutator hard to grasp. Is it a computation device or does it have an intuitive geometric meaning? Can anyone give me an example of two non-commutative vector fields? Thanks!

I'm trying to use LaTeX to graph both the vectors of the electric field around a dipole and the field lines. So far I have a quiver plot of the vector field: I obtained this by using the code \begin{tikzpicture} \def \U{(x-1)/((x-1)^2+y^2)^(3/2) - (x+1)/((x+1)^2+y^2)^(3/2)} \def...

Homework Statement I am trying to show that Majorana vector current vanishes. I am following this article and I am trying to get to the very right hand side of eq. (27). Homework Equations \psi_M^C = \psi_M,\\ \psi^C_M = C \overline{\psi}_M^T,\\ C^T=-C, \hspace{1cm} C^T\gamma_{\mu}C =...

I want to visualize the concept of divergence of a vector field.I also have searched the web.Some says it is 1.the amount of flux per unit volume in a region around some point 2.Divergence of vector quantity indicates how much the vector spreads out from the certain point.(is a...

Could there be a connection between Robert Zimmermann's work (McMaster Univ. Toronto) on Vector Plasma, and Jenkins and Fischbach's (Perdue Univ.) work on variations in the rate of radioactive decay for elements on Earth in relation to solar activity? Only looking for a confirmation that their...

I am learning the basics of differential geometry and I came across tangent vectors. Let's say we have a manifold M and we consider a point p in M. A tangent vector ##X## at p is an element of ##T_pM## and if ##\frac{\partial}{\partial x^ \mu}## is a basis of ##T_pM##, then we can write $$X =...

Given a basis of a vector space $(V,O_1,O_2)$ can it represent two different non-isomorphic graphs.Any other inputs kind help. It will improve my knowledge way of my thinking. Another kind help with this question is suppose (V,O_1,O_2) and (V,a_1,a_2) are two different vector spaces on the...

When comparing Newtonian and GR views of gravity, I came across a vector expression in the Newtonian form which happens to integrate to the total potential energy of a system of masses, even in the case of dynamic situations: ##-\mathbf{x}\cdot\rho \, \mathbf{g}##, where ##\mathbf{x}## is...

Hello all! I've got this problem I'm trying to do, but I'm not sure what the best way to approach it is. It's obvious that there can only be 2 dimensions, because there's only two linearly independent vectors in the span. However, what would be a good way of using the inequalities to prove...

Homework Statement A vector n of magnitude 8 units is inclined to x,y and z axis at 45, 60 and 60 degrees resoectively.If the plane passes through (root2, -1, 1) and is normal to n then find its equation. Homework Equations (r-a).n=0 where r is position vector of a point on plane, a is a point...

The theorem is as follows: All finite dimensional vector spaces of the same dimension are isomorphic Attempt: If T is a linear map defined as : T : V →W : dim(V) = dim(W) = x < ∞ & V,W are vector spaces It would be sufficient to prove T is a bijective linear map: let W := {wi}ni like wise let...

I'm given equations of surfaces and asked for the vector function that represents the intersection of the two surfaces. For ex: $$x^2 + y^2 = 4$$ and $$z = xy$$ In the solutions manual the answer is given like this: a sum of terms of cos t and sin t (is this polar form?). The way I did wasn't...

I seem to have come across a new problem I am trying to programm a sloping pipe system to match an array of vector points. I thought I had it all sorted out until I tested with a very high slope angle. With a normal slope of around 2% everything looks fine. If I increase the slope to 30% there...

I know how to find the cos(theta) between two vectors but I do not know how to find the sin(theta). vector w=i+3j vector v=<5, 2>

I am reading Miroslav Lovric's book: Vector Calculus ... and am currently focused n Section 1.3: The Dot Product ... I need help with an apparently simple matter involving Theorem 1.6 and the section on the orthogonal vector projection and the scalar projection ...My question is as follows: It...

I have a vector ##\textbf{v} \in \mathbb{R}^{3N}## and a function ##\textbf{Ψ} : \mathbb{R}^{3N} \longrightarrow \mathbb{R}^p## such that ##\textbf{Ψ}(\textbf{v})=0##. Why the set ##T=\{ \textbf{x} \in \mathbb{R}^{3N} \ | \ \textbf{Ψ}(\textbf{x})=0 \}## has dimension ##n=3N-p##?

Hi. A beam of previously unpolarized or diagonally polarized doesn't create an interference pattern behind a double slit if there is a vertically and horizontally oriented polarizer behind either slit. The classical explanation is that the electric field is a vector perpendicular to the...

Homework Statement how to write the vector equation of the line of shortest distance between two skew lines in the shortest and most efficient way? (The exact lines given in a particular problem in my book can be referenced- L1=(3i+8j+3k)+λ(3i-j+k) and L2=(-3i-7j+6k)+μ(-3i+2j+4k) ) 2. Relevant...

Homework Statement A vector r has length 21 and direction ratio's 2,-3,6. The direction cosines of r, given that r makes an obtuse angle with x-axis is given by? Homework Equations l/a = m/b =n/c ...(1) (l,m,n are direction cosines, a,b,c are direction ratios l^2 + m^2 + n^2=1...(2) The...

Homework Statement Find the scalar, vector, and parametric equations of a plane that has a normal vector n=(3,-4,6) and passes through point P(9,2,-5) Homework EquationsThe Attempt at a Solution Finding the scalar equation: Ax+By+Cz+D=0 3x-4y+6z+D=0 3(9)-4(2)+6(-5)+D=0 -11+D=0 D=11...

Homework Statement A scientist investigating the movements of dolphins in the Mediterranean uses a dart gun to shoot small, harmless tracking devices onto the fins of dolphins. When standing on deck, her hand is 1m above the water, and looking along the dart gun she is holding at an angle of...

Homework Statement [/B] Write vector and parametric equations for the line that goes through the points P(–3, 5, 2) and Q(2, 7, 1). Homework EquationsThe Attempt at a Solution First I find the direction vector for PQ. PQ=Q-P = (2,7,1)-(-3,5,2) =[2-(-3),7-5,1-2] =5,2,-1 PQ= (5,2,-1) Now I...

Homework Statement (Problems/diagrams referenced are attached as images.) Homework Equations Net torque about an origin = time derivative of the angular momentum vector about the same origin. The Attempt at a Solution I've solved these problems before, but I'm now looking back at them and...

Let ##\vec { A }## = ##a \dot { i } + b \hat { j } + c \hat { k }## My question is "is ##\frac { 1 } { \vec { A } }## is a vector or not and if yes then what is it's components?"

since it is known that ##\vec{A_\perp} = -{mG \over R^2}## why did the professor write it as ##\vec{A_\perp} = {- R G \rho \over 3}## for perfect sphere with perfect mass distribution ? Shouldn't it be ##\vec{A_\perp} = -{4 \over 3} \pi R G \rho##? I need help thanks.

Homework Statement A rocket is to rendezvous with a satellite and needs to make a course adjustment. the rocket has a velocity = (10 + 0 + 0) ms−1 relative to the satellite and mission control has sent a command to the rocket side thruster to exert a thrust = (0 − 100 + 0) N for 100 seconds...

Is the equation presented (that the time-derivative of a given vector in such a scenario is equal to its angular frequency vector cross the vector itself) true in the case of a vector whose origin is not on the axis of rotation? The way I'm visualizing this, if we take such a displaced origin...

Is there a sensible way of defining a displacement vector in a general manifold? That is, the displacement vector being the difference between position vector at two different points... the problem is that these two different points have, in general, different tangent vector spaces. Never the...

Hello all, Im trying to do a simulation of a poynting vector of an electromagnetic wave and I assume the following: At t=0 the E-field vector is (0,0,e^(-ikx)) and the H-field vector (0,e^(-ikx),0), hence orthogonal to it in vaccum, which is assumed, also the amplitudes are simplified both to...

Homework Statement For a vector field $$\begin{equation} X:=y\frac{\partial{}}{\partial{x}} + x\frac{\partial{}}{\partial{y}} \end{equation}$$ Find it's integral curves and the curve that intersects point $$p = \left(1, 0 \right).$$ Show that $$X(x,y)$$ is tangent to the family of curves: $$x^2...

Consider ##X## and ##Y## two vector fields on ##M ##. Fix ##x## a point in ##M## , and consider the integral curve of ##X## passing through ##x## . This integral curve is given by the local flow of ##X## , denoted ##\phi _ { t } ( p ) .## Now consider $$t \mapsto a _ { t } \left( \phi _ { t } (...

We all know that the area of a triangle having consecutive sides as ##\vec { a }## and ##\vec { b }## has the area ##\frac { 1 } { 2 } | \vec { a } \times \vec { b } |## but what is the direction of that area vector? I mean if we consider ##\vec { a } \times \vec { b }## that will be one...

Homework Statement I am studying co- and contra- variant vectors and I found the video at youtube.com/watch?v=8vBfTyBPu-4 very useful. It discusses the slanted coordinate system above where the X, Y axes are at an angle of α. One can get the components of v either by dropping perpendiculars...

Is there a way of subtracting two vectors in spherical coordinate system without first having to convert them to Cartesian or other forms? Since I have already searched and found the difference between Two Vectors in Spherical Coordinates as...

Hey! :o We consider the $\mathbb{F}_2$-vector space $(2^M, +, \cap)$, where $M$ is non-empty set and $+ : 2^M\times 2^M \rightarrow 2^M: (X,Y)\mapsto (X\cup Y)\setminus (X\cap Y)$. I want to show that $(2^M, +, \cap )$ for $\mathbb{K}=\{\emptyset , M\}$ satisfies the axioms of a vector space...