What is Position vector: Definition and 110 Discussions
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point P in space in relation to an arbitrary reference origin O. Usually denoted x, r, or s, it corresponds to the straight line segment from O to P.
In other words, it is the displacement or translation that maps the origin to P:
r
=
O
P
→
{\displaystyle \mathbf {r} ={\overrightarrow {OP}}}
The term "position vector" is used mostly in the fields of differential geometry, mechanics and occasionally vector calculus.
Frequently this is used in two-dimensional or three-dimensional space, but can be easily generalized to Euclidean spaces and affine spaces of any dimension.
I think I have completed the exercise but since I have seldom used polar coordinates I would be grateful if someone would check out my work and tell me if I have done everything correctly. Thanks.
My solution follows.
Since ##\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2=1## it follows...
First question:
Does the position vector exist in the tangential and normal coordinate system in curvilinear motion?
↧↧↧
Regarding my teacher's response, he confirmed the existence of the position vector formula but was unable to provide it. Upon returning home, I attempted to find its formula...
Suppose I have a Cartesian Coordinate system (x,y) and a polar coordinate system (##r, \theta##). The position vector (3,4) and (5, ##\arctan \frac{4}{3}##) are the same except the representation. The position vector is a tensor, how does the position vector follow the tensor transformation...
To derive ##\vec r (t)=(−Rsin(ωt),Rcos(ωt)) ##
I start by integrating ##ω=\frac{dθ}{dt}## to get ##θ_f=θ_i+ωt##.
Therefore since ##Δθ=θ## by definition since the angular displacement is always taken with respect to some initial reference line, then ##θ_f−θ_i=θ## , thus, ##\theta = \omega t##...
Is the position vector a real vector?
I have a hard time with this question because vectors are unchanged if I were to change my reference frame.
Example: If I place a pencil such that it points towards the door. It doesn’t matter what I define my origin to be. The pencil’s length and direction...
Highlighted part only...
Part (a) was easy ##2\sqrt 5##.
For part (b),
...##BC=4i+2j##
it follows that,
##OC=OB+BC##
##OC=3i+5j+4i+2j=7i+7j## correct? any other better approach guys!
For part (c),
I will form the equations as follows;
Let ##D(x,y)## then,
##x-4=2(4-3)##
and...
for 3di i did the normal AB=BC so b-a would give either satisfy or not this phenomenon, my answer was (3a-1, -4) & (2a^2 + a - 1, 4a - 2), now how would i know from here if they're collinear or not?
Good Morning!
I understand that a vector is a physical object
I understand that it is the underlying basis that determines how the components transform.
However, I encounter this:
https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors
The fifth paragraph has this statement
A...
what would be the y'-x' ##\vec r## vector be?
I think it is
##\vec r = (8t - 1) \hat i + (6t - 2) \hat j## (not sure whether it is correct or not.)
I thought about it as at t = 0 the position needs to be -1i -2j so that is why I took the signs in the y'-x' frame position vector as a - instead...
I had an equation. $$T=\frac{1}{2}m[\dot{x}^2+(r\dot{\theta})^2]$$ Then, they wrote that $$\mathrm dr=\hat r \mathrm dr + r \hat \theta \mathrm d \theta + \hat k \mathrm dz$$ I was thinking how they had derived it. The equation is looking like, they had differentiate "something". Is it just an...
It is said that: It is not possible to write a position vector in a curved space time. What is the reason?
How can one describe a general vector in a curved space time?
Can you please suggest a good textbook or an article which explains this aspect?
1.)##\dot{\vec{r}}=\dot{x}\hat{i}+\dot{y}\hat{j}+\dot{z}\hat{k}=\dot{r}\hat{r}## since the unit vector is constant
2.) ##\dot{r}\hat{r}=\frac{x\hat{i}+y\hat{j}+z\hat{k}}{\sqrt{x^2+y^2+z^2}}\frac{\dot{x}x+\dot{y}y+\dot{z}z}{\sqrt{x^2+y^2+z^2}}##...
I have not tried to make any calculation. It's nonsense, because I don't understand the statement. The first vector points to the west. Given a two dimensional coordinate system, the first vector is pointing to the left. I imagine geographical coordinates, north (+y), south (-y), west (-x), and...
My solution is making an analogy of the ##\text{Relevant equations}## as shown above, starting from the equation ##\vec \omega = \frac{1}{2} \vec \nabla \times \vec v##.
We have ##\vec B = \vec \nabla \times \vec A = \frac{1}{2} \vec \nabla \times 2\vec A \Rightarrow 2\vec A = \vec B \times...
I know the divergence of any position vectors in spherical coordinates is just simply 3, which represents their dimension. But there's a little thing that confuses me.
The vector field of A is written as follows,
,
and the divergence of a vector field A in spherical coordinates are written as...
Homework Statement
I'm a bit confused on when a force creates a moment about a point, and when it does not. In particular, in the attached diagram, would F be able to produce a moment about point A? I initially thought that wouldn't be possible as A lies on the same line as F, but since F does...
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...
Homework Statement
a particle's position is the vector r=(ct^2-2dt^3)i+(4ct^2-dt^3)j where c and d are positive constants. find the expression for the x-component of the velocity (for time t>0) when the particle is moving in the x-direction. you should express your answer in terms of variables...
Hi, I've got this problem I'm trying to work out. The problem seems simple, but I don't think I have a good way to construct a way to solve it.
This is the problem.
Let P and Q be two points with position vectors p and q and let
R be a point midway between these two. Find an expression for
the...
What is the covariant derivative of the position vector $\vec R$ in a general coordinate system?
In which cases it is the same as the partial derivative ?
Homework Statement
The position vector of an object of mass 0.10 kg at time t in seconds is given by
=(^3+5)−4+2^2
Find the velocity and the acceleration as a function of t
Homework Equations
=(^3+5)−4+2^2
The Attempt at a Solution
For velocity I think the equation needs to be...
Homework Statement
I am trying to solve for change in velocity for the center of a rim with respect to the contact patch of a tire that has some degree of camber. The equation finalized is shown in the image below, equation 2.6.
http://imgur.com/a/oHucp
Homework EquationsThe Attempt at a...
Homework Statement
Bead on spoke:
constant speed ##u## along spoke
it starts at center at ##t=0##
angular position is given by ##\theta=\omega t##, where ##\omega## is a constant
Homework Equations
## \frac{d\hat r}{dt} = \dot \theta \hat \theta ## (1)
## \frac{d\hat \theta}{dt} = -\dot...
Homework Statement
My homework problem is a proof in orbital mechanics, but I'm not looking for specific help on that just yet, I'd like to work through it on my own. In doing so however, I'm having a hard time conceptualizing the idea of derivatives of vectors at a specified time. If r is a...
I have read that in polar coordinates, we can form the position vector, velocity, and acceleration, just as with Cartesian coordinates. The position vector in Cartesian coordinates is ##\vec{r} = r_x \hat{i} + r_y \hat{j}##. And any choice of ##r_x## and ##r_y## maps the vector to a position in...
While proving the Midpoints of the Sides of a Quadrilateral Form a Parallelogram , I got bogged down with position vectors.
Let a,b,c and d be the position vectors of A,B,C and D. But where is the origin? Aren't we supposed to locate position of origin?
Homework Statement
The position vector of a particle at time t is R=(1-t^2)i+(3t-5t^2)j. Find the time at which P is moving (a) towards the origin (b) away from the origin.[/B]
Homework EquationsThe Attempt at a Solution
I've thought about this for a while but I've come to the conclusion...
Given the definition of the covariant basis (##Z_{i}##) as follows:
$$Z_{i} = \frac{\delta \textbf{R}}{\delta Z^{i}}$$
Then, the derivative of the covariant basis is as follows:
$$\frac{\delta Z_{i}}{\delta Z^{j}} = \frac{\delta^2 \textbf{R}}{\delta Z^{i} \delta Z^{j}}$$
Which is also equal...
The equation following (3.80) seems to suggest that the velocity vector ##\vec{\dot{r}}## must always be parallel to the position vector ##\vec{r}##. But clearly this is not true as a particle's velocity can be in any direction.
What's wrong?
Let $P = (1,2,-1)$ and $Q = (3,1,0)$. A point $B$ lies on the line segment between $P$ and $Q$. Given that its distance from $P$ is twice its distance from $Q$. Find the position vector of $B$.
Hi guys,
I got one confusion when reading Goldstein's Classical Mechanics (page 20, third edition). After getting the equation,
then it says that
Note that in a system of Cartesian coordinates the partial derivative of T with respect to qj vanishes. Thus, speaking in the language of...
Homework Statement
The acceleration of a particle moving only on a horizontal xy plane is given by a = 3ti + 4tj, where a is in meters per second squared and t is in seconds. At t = 0, the position vector r = (20.0 m)i + (40.0 m)j locates the particle, which then has the velocity vector v =...
Hello,
I need to create a 2-D electron energy density plot in Mathematica to compare with my STM experimental results in my lab class. This would be done by plotting the superposition of the symmetric and anti-symmetric wave functions,
$$\Psi_s(\textbf{r}) =...
Torque is defined as the cross product of position vector and force, i.e. \vec \tau = \vec r \times \vec F .
However the force vector \vec F is fixed, but the choice of origin is arbitrary, making \vec r also arbitrary. Does it make the torque vector also arbitrary, which apparently shouldn't...
Homework Statement
A particle moves in a plane described by the position vector r (t)= ( 2bsin( wt ))i + ( bcos( wt ) )j
where b and w are some constants. The angle between its velocity and acceleration vectors at time t=(π/2w)
a) is approximately 27 degree .
b) is exactly 45 degree .
c) is...
Homework Statement
I have somewhat general question about time derivative of a vector.
If we have
r=at2+b3
it's easy to find instantaneous acceleration and velocity(derivative with respect to dt)
v=2at+3bt2
a=2a+6bt
But consider this position vector
r=b(at-t2)
where b is constant vector and a...
Homework Statement
2. The attempt at a solution
I would use the arctan (Position j / Position i) and set it equal to the position vector. Then I would substitute values from the graph to find the variables.
The solution manual, however, takes the derivative of the position vector and then...
A particle P is moving with a const. speed of 6m/s in a direction 2î - j - 2k. When t=0, P is at a point with position vector 3i + 4 j -7k. Find the position vector of P after (1) t seconds
(2) 4 seconds.
The solutionnstates that ' change the particle speed is constant and is a long text...
The position vector ##\vec{r}## in cartesian coordinates is: ##\vec{r} = x \hat{x} + y \hat{y}##, in polar coordinates is: ##\vec{r} = r \hat{r}##. But, given a curve s in somewhere of plane, with tangent unit vector ##\hat{t}## and normal unit vector ##\hat{n}## along of s, exist a definition...
What is a position vector? Is their any difference between the position vector and position? Isnt position of a point supposed to represent its direction in Cartesian plane as well(Positive quadrants , negative quadrants). So why two different terms?
Homework Statement
The distance traveled by an airplane flying from San Francisco International airport (SFO) to San Jose International (SJC) is 30 nautical miles 36° south of east. Flying from SJC to Tracy, the plane's displacement is 36 nautical miles 52° east of north. What is Tracy's...
Homework Statement
An object has a position given by r-> = [2.0 m + (1.00 m/s)t] i + [3.0m−(5.00 m/s2)t2] j, where all quantities are in SI units. What is the magnitude of the acceleration of the object at time t = 2.00 s?
Homework Equations
All I am thinking here is that I can find...
I'm given the position vector as a function of time for a particle (b, c and ω are constants):
\vec{r(t)} = \hat{x} b \cos(ωt) + \hat{y} c \sin(ωt)
To obtain it's velocity i differentiate \vec{r(t)} with respect to time and i obtain:
\vec{v(t)} = -\hat{x} ωb \sin(ωt) + \hat{y} ωc...
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...