Geometric object with magnitude (length) and direction.
13
votes
4answers
4k views
How can area be a vector?
My professor told me recently that Area is a vector. A Google search gave me the following definition for a vector:
Noun: A quantity having direction as well as magnitude, esp. as
determining ...
10
votes
3answers
352 views
Representing forces as one-forms
First of all, sorry if any of those things are silly or nonsense, I'm just trying to understand better how the concepts of forms, exterior derivative and so on can be used in physics.
This question ...
9
votes
5answers
936 views
How is it that angular velocities are vectors, while rotations aren't?
Does anyone have an intuitive explanation of why this is the case?
8
votes
6answers
9k views
What is the physical significance of dot & cross product of vectors? Why is division not defined for vectors?
I get the physical significance of vector addition & subtraction. But I don't understand what do dot & cross products mean?
More specifically,
why is it that dot product of vectors ...
8
votes
4answers
573 views
Is a 1D vector also a scalar?
A vector in one dimension has only one component. Can we consider it as a scalar at the same time?
Why time is not a vector, although it can be negative and positive (when solving for time the ...
7
votes
6answers
817 views
Quaternions and 4-vectors
I recently realised that quaternions could be used to write intervals or norms of vectors in special relativity:
$(t,ix,jy,kz)^2 = t^2 + (ix)^2 + (jy)^2 + (kz)^2 = t^2 - x^2 - y^2 - z^2$
Is it ...
7
votes
6answers
260 views
In coordinate-free relativity, how do we define a vector?
Relativity can be developed without coordinates: Laurent 1994 (SR), Winitzski 2007 (GR).
I would normally define a vector by its transformation properties: it's something whose components change ...
6
votes
3answers
1k views
Physics of a skateboard ollie
Does anyone have a good explanation of the physics and vectors of force involved in the skateboarding trick the ollie (where the skater jumps and causes the skateboard to rise off the ground with ...
5
votes
6answers
2k views
How is gradient the maximum rate of change of a function?
Recently I read a book which described about gradient. It says
$${\rm d}T~=~ \nabla T \cdot {\rm d}{\bf r},$$
and suddenly they concluded that $\nabla T$ is the maximum rate of change of $f(T)$ ...
5
votes
1answer
227 views
What does scalar phi represent in spacetime?
Trying to understand one-forms and vectors via Schutz's A First Course In General Relativity.
His example uses a spacetime diagram, a scalar field phi, a curve (worldline) parametrized using proper ...
5
votes
1answer
128 views
Vector and Spinor Representation in Ramond-Neveu-Schwarz Superstring Theory
I am learning Ramnond-Neveu-Schwarz Superstring theory (RNS theory). I often find the following notation, especially in the closed string spectrum etc.:
$$\mathbf{8}_s,\mathbf{8}_v $$
And it is ...
5
votes
4answers
298 views
Are covariant vectors representable as row vectors and contravariant as column vectors
I would like to know what are the range of validity of the following statement:
Covariant vectors are representable as row vectors. Contravariant
vectors are representable as column vectors.
...
4
votes
3answers
254 views
Meaning of the direction of the cross product
I was doing calculations with torque and then I came across something very confusing:
I understand that the magnitude of the torque is given by product of the displacement(from the center of ...
4
votes
1answer
1k views
Uniqueness of Helmholtz decomposition?
Helmholtz theorem states that given a smooth vector field $\pmb{H}$, there are a scalar field $\phi$ and a vector field $\pmb{G}$ such that
$$\pmb{H}=\pmb{\nabla} \phi +\pmb{\nabla} \times \pmb{G},$$
...
4
votes
2answers
115 views
Vector potential
I have difficulty understanding the following vector calculus example. Text can be found here. It is the 5th Q&A -- starting with equation (31.1035).It concerns finding the vector potential of a ...
4
votes
2answers
83 views
Why distinguish between row and column vectors?
Mathematically, a vector is an element of a vector space. Sometimes, it's just an n-tuple $(a,b,c)$. In physics, one often demands that the tuple has certain transformation properties to be called a ...
3
votes
2answers
331 views
Is (0,0,0) an undefined vector?
I'm not sure what to make of the direction of a vector with components (0,0,0). Is it an undefined vector?
3
votes
4answers
203 views
How to apply an algebraic operator expression to a ket found in Dirac's QM book?
I've been trying to learn quantum mechanics from a formal point of view, so I picked up Dirac's book. In the fourth edition, 33rd page, starting from this:$$\xi|\xi'\rangle=\xi'|\xi'\rangle$$
(Where ...
3
votes
2answers
70 views
Sign of acceleration
I'm developing an application using accelerometer sensor. I'm not good at physics so forgive me if the question is trivial. If I have 3 values of acceleration: $x$, $y$, $z$, I find acceleration ...
3
votes
2answers
146 views
Is length/distance a vector?
I have heard that area is a vector quantity in 3 dimensions, e.g. this Phys.SE post, what about the length/distance? Since area is the product of two lengths, does this mean that length is also a ...
3
votes
1answer
60 views
Electric field a distance $z$ above the center of a circular loop. The Hard way [closed]
Problem 2.5: Find the electric field a distance $z$ above the center of a circular loop of radius $r$ which carries a uniform line charge $\lambda$.
This problem is in refereced here (with ...
3
votes
0answers
72 views
Uniqueness of the vector in $\mathbb{R}^n$ specified by the curl, divergence and the normal component [migrated]
If I know the curl, and divergence of a n-component vector in a region, and its normal component around its boundary, is the vector uniquely specified? If yes, how do I prove it? Also, is there a ...
2
votes
3answers
22k views
Linear acceleration vs angular acceleration equation
I'm learning about angular velocity, momentum, etc. and how all the equations are parallel to linear equations such as velocity or momentum. However, I'm having trouble comparing angular acceleration ...
2
votes
3answers
240 views
How to distinguish 4D and 3D vectors in handwriting?
Usually vectors are denoted with bold font in printbooks and with arrows above in handwriting.
In Thorn's e al. Gravitation, 4D vectors are denoted with bold and 3D vectors with bold italic. How to ...
2
votes
2answers
426 views
Nature of spacetime 4-vector and tangent space?
An entry level confusion about spacetime. I understand that a 4-vector describes a point or event in spacetime. But I've also read (Bertschinger, 1999) that re spacetime "we are discussing tangent ...
2
votes
2answers
1k views
split gravitational force into x, y, and z componenets
I am writing a program for a computer science class in which I am doing an n-body simulation in 3-dimensional space. Currently, I have figured out the gravitational force along the hypotenuse between ...
2
votes
2answers
188 views
Coordinate transformation from earth to solar
I am building a 3d model of the solar system and need to figure out the position of the pole stars of each planet in order to tilt the planets in the correct direction the correct amount. I've already ...
2
votes
2answers
472 views
Meaning of angular velocity in a rotating system
When you study the motion of a rigid body you have $\vec\omega$, the vector associated to angular velocity. In the case you are using Euler angles and want a quick formula for the rotational kinetic ...
2
votes
1answer
502 views
Which are other anomalies like Divergence of 1/r^2?
As one might have learned in the basic science (ex. Electrodynamic
theory), when we apply the divergence theorem to the vector function
like 1/r^2 with it pointing in the radial direction (like ...
2
votes
3answers
445 views
negative vectors (eg velocity)
If you said someone had a velocity of -12mph and they were traveling north? Wouldn't it mean that they were traveling 12mph south?
This is a quote from here:
if something [object-x] moving to the ...
2
votes
2answers
71 views
Sum of all forces
Let us glue up these two images, where we get closed loop thrust of water. Force $F_3$ has direction $-x$ and force $F_2$ has $x$ direction. What is the sum of all forces? Can it be more than zero? ...
2
votes
1answer
133 views
Vector transformation in special relativity
Please note that I am very new on this website so have some difficulties in writings as required here but trying really hard to learn quickly. La-Tex is the main problem but please understand me that ...
2
votes
2answers
62 views
Another question about Shankar's notation
I have another question on the notation in Shankar. I think it's sloppy, but I also may just be misunderstanding it. Again, this is at the very beginning of the math intro.
He has:
$$a\left| V ...
2
votes
1answer
1k views
How Does force relate to velocity
I had originally asked this question on math overflow and it was suggested that I ask it here.
So I know that a force will change the magnitude of velocity if it is at an angle other that 90 degrees. ...
2
votes
3answers
137 views
What is the general approach to calculating time of impact in 3D?
Given two objects a and b moving at fixed velocities, how would you determine (a) whether they will collide at all, and if so, (b) time of impact?
(Let us assume these are spherical bodies each with ...
2
votes
1answer
308 views
Combined Gravitational Force Vectors in a Spherical Coordinate System
Asking a question here is quite intimidating for me -- while I love Physics, my high-school understanding only allows me to go so far... I've been trying to solve this problem for the last couple of ...
2
votes
1answer
147 views
Why can we use just one angular velocity vector to describe the rotation of a whole non-inertial reference frame?
The other day in class the professor was explaining non-inertial reference frames. We were working out how to find the acceleration of a point as measured from the non-inertial reference frame, and ...
2
votes
0answers
43 views
Special case of the hodge decomposition theorem [migrated]
I am trying to prove the following special case of the hodge decomposition theorem in differential geometry for a n component vector field $V_i$ in $\mathbb{R}^n$.
any vector can be written as the ...
2
votes
2answers
309 views
2D - Kinematics - Linkage System using Vector Algebra
I have this question that I dont know how to solve correctly :
My question is, how do I find $V_B$ ? I will find the angular velocities myself, but I want to know the method to get $V_B$ ?
I know ...
2
votes
0answers
76 views
Surface normal on the earth to the sun at a given point in time
How complicated is it to calculate a surface normal on the spherical approximation of the earths surface pointing towards the sun at a given point in time?
What I try do is to highlight a small area ...
1
vote
3answers
6k views
Uses of Vectors In Real Life
I always wonder how vectors are used in real life.Vectors and decomposition of vectors,dot and cross products are taught in the early stage in every undergraduate physics course and in every ...
1
vote
3answers
932 views
Can vectors in physics be represented by complex numbers and can they be divided? [closed]
Below is attached for reference, but the question is simply about whether vectors used in physics in a vector space can be represented by complex numbers and whether they can be divided.
In ...
1
vote
6answers
131 views
What is the direction of a point vector. A vector with magnitude 0? [closed]
A simple question :
Can a point on a piece of paper represent a vector ? Can i say that a point "B" ( magnitude =0, because it's a point) , is having direction towards +x axis ?
Thanks
1
vote
2answers
70 views
Angular displacement and the displacement vector
I've been recently introduced to angular displacement, and I'm a little bit confused about it. I think that displacement which is a vector and which is defined as the shortest distance between any two ...
1
vote
2answers
87 views
Points in Spacetime
Assume there are two points in spacetime $a=(t,x,y,z)$ and $a'=(t',x',y',z')$. Let's say that the first one is in the origin of spacetime i.e. $a=(0,0,0,0)$. The point $a'$ has two possibilities
...
1
vote
2answers
163 views
Conservation of Linear Momentum with respect to a given direction
Is linear momentum conserved in any direction? More specifically, if you project all momentum vectors in a system onto another vector, will momentum be conserved?
I know that momentum is conserved ...
1
vote
2answers
209 views
Question on notation in Shankar's Quantum Mechanics - math intro on vector spaces
I'm just beginning Shankar's 2nd edition Quantum Mechanics and having some trouble with notation. He defines his vectors as "$\left|V\right>$" . And with a scalar multiplier as "$a\left|V\right>$" . ...
1
vote
2answers
160 views
Why no basis vector in Newtonian gravitational vector field?
In my textbook, the gravitational field is given by$$\mathbf{g}\left(\mathbf{r}\right)=-G\frac{M}{\left|\mathbf{r}\right|^{2}}e_{r}$$
which is a vector field. On the same page, it is also given as a ...
1
vote
1answer
1k views
The resultant of two forces acting at any angle?
I am studying about forces as vectors. And they give me this equation:
$c^2 = a^2 + b^2 - 2ab \cos C$
Can anybody explain me the second part of the equation? I perfectly understand $c^2 = a^2 + b^2$ ...
1
vote
1answer
77 views
rotation matrix - why am I thinking this wrong?
The rotation given in Question 1 part ii) doesn't match with this wikipedia link http://en.wikipedia.org/wiki/Rotation_matrix.
$$ \begin{array}{lcl}
x' &=& x \cos\theta - y \sin\theta \\
y' ...
