Questions tagged [vectors]

Geometric object with magnitude (length) and direction.

Filter by
Sorted by
Tagged with
71
votes
19answers
28k views

Is weight a scalar or a vector?

My professor insists that weight is a scalar. I sent him an email explaining why it's a vector, I even sent him a source from NASA clearly labeling weight as a vector. Every other source also ...
59
votes
8answers
109k 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 $\vec{A}...
37
votes
8answers
4k views

Is it foolish to distinguish between covariant and contravariant vectors?

A vector space is a set whose elements satisfy certain axioms. Now there are physical entities that satisfy these properties, which may not be arrows. A co-ordinate transformation is linear map from a ...
36
votes
5answers
3k views

When is it useful to distinguish between vectors and pseudovectors in experimental & theoretical physics?

My understanding of pseudovectors vs vectors is pretty basic. Both transform in the same way under a rotation, but differently upon reflection. I might even be able to summarize that using an equation,...
30
votes
5answers
49k 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 the ...
24
votes
9answers
8k views

How to interpret the units of the dot or cross product of two vectors?

Suppose I have two vectors $a=\left(1,2,3\right)$ and $b=\left(4,5,6\right)$, both in meters. If I take their dot product with the algebraic definition, I get this: $$a \cdot b = 1\mathrm m \cdot 4\...
24
votes
6answers
80k views

Can we divide two vectors?

Can we divide two vector quantities? For eg., Pressure( a scalar) equals force (a vector) divided by area (a vector).
23
votes
5answers
107k views

Why is current a scalar quantity?

Current has both magnitude and direction. As per the definition of vector defined in encyclopedia, current should be a vector quantity. But, we know that current is a scalar quantity. What is the ...
23
votes
3answers
2k views

Representing forces as one-forms

This question arose because of my first question Interpreting Vector fields as Derivations on Physics. The point here is: if some force $F$ is conservative, then there's some scalar field $U$ which is ...
22
votes
8answers
4k views

Mass and Newton's Second Law

While trying to understand the second law of Newton from "An Introduction to Mechanics" by Kleppner and Kolenkow, I came across the following lines that I don't understand: "It is natural to ...
19
votes
12answers
8k views

Why is force a vector?

"We have focused our discussion on one-dimensional motion. It is natural to assume that for three-dimensional motion, force, like acceleration, behaves like a vector."- (Introduction to Mechanics) ...
18
votes
4answers
8k views

Why does torque point perpendicular to direction of the motion?

I have an intuition problem calculating torque using the cross product formula. As for example let the magnitude of the force be 50 lbs and length of the wrench be one foot and you are exerting ...
18
votes
2answers
2k views

Is the right hand rule a trick to avoid tensors?

I have read in this answer that "to represent angular momentum as a vector you need to use a right hand rule. This is annoying, because physics at ordinary scales is reflection invariant but ...
18
votes
4answers
8k 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 ...
17
votes
7answers
6k 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?
17
votes
9answers
4k 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 ...
16
votes
8answers
3k views

Formal Definition of Dot Product

In most textbooks, dot product between two vectors is defined as: $$\langle x_1,x_2,x_3\rangle \cdot \langle y_1,y_2,y_3\rangle = x_1 y_1 + x_2 y_2 + x_3 y _3$$ I understand how this definition ...
16
votes
4answers
2k views

Does spacetime position not form a four-vector?

When one starts learning about physics, vectors are presented as mathematical quantities in space which have a direction and a magnitude. This geometric point of view has encoded in it the idea that ...
16
votes
2answers
6k views

Why add a minus sign in the formula for gravity?

Why do we add a minus sign in our formula for gravity, when we might as well choose the unit vector $r_{21}$, instead of $r_{12}$? I'm just wondering why we choose this convention. Is it because it'...
15
votes
5answers
1k views

What does it mean for a physical quantity if its mixed second partial derivatives are not equal?

This goes for every problem (either in electromagnetism or fluid dynamics) that has to do with vector fields. Say we have a fluid flowing in a closed circular pipe (or an electromagnetic field, the ...
14
votes
6answers
2k 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 ...
13
votes
4answers
2k views

Neither a vector, nor a scalar

While I was reading a book on mechanics, when introducing the vector multiplication the author stated that multiplying two vectors can produce a vector, a scalar, or some other quantity. 1.4 ...
13
votes
3answers
23k views

Direction of angular velocity

Angular velocity is the rate of angular displacement about an axis. Its direction is determined by right hand rule. According to right hand rule, if you hold the axis with your right hand and rotate ...
13
votes
5answers
9k views

Why can the cross product of two vectors be calculated as the determinant of a matrix?

The cross product $\vec{a} \times \vec{b}$ can be written as the determinant of the matrix: $$\left| \begin{matrix} \vec{i} & \vec{j} & \vec{k} \\ a_i & a_j & a_k \\ b_i & b_j &...
13
votes
4answers
3k 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. For ...
12
votes
5answers
11k views

If force is a vector, then why is pressure a scalar? [duplicate]

By definition pressure is the perpendicular force applied to a unit area. So it has a direction which is perpendicular to the area. So it should be a vector. But I did sone googling and found out that ...
12
votes
4answers
3k views

Why are angular mometum and angular velocity not necessarily parallel, but linear momentum and linear velocity are always parallel?

I have read that it's not necessary for angular momentum and angular velocity to be parallel, but it is necessary for linear momentum and linear velocity to be parallel. How is this correct?
12
votes
5answers
3k views

Does the magnitude of a physical quantity have units or is it just a plain number?

Does the magnitude of a physical quantity have units? For example, if a velocity vector is $36\ \mathrm{m\,s^{-1}}\ \hat{u}$, is its magnitude $36\ \mathrm{m\,s^{-1}}$ or just $36$? Also why?
12
votes
7answers
1k views

Can we divide a vector by another vector? How about this: $a = vdv/dx?$

My physics teacher told us that we can’t divide vectors, that vector division has no physical meaning or significance. How about this: $$a = vdv/dx.$$ It says acceleration vector equals velocity (as ...
12
votes
2answers
3k views

What does it mean to transform as a scalar or vector?

I'm working through an introductory electrodynamics text (Griffiths), and I encountered a pair of questions asking me to show that: the divergence transforms as a scalar under rotations the ...
12
votes
6answers
1k views

How do we prove that the 4-current $j^\mu$ transforms like $x^\mu$ under Lorentz transformation?

Given that the position vector $\textbf{r}$ to be a vector under rotation, we mean that it transforms under rotation as $\textbf{r}^\prime=\mathbb{R}\textbf{r}$. Now, taking two time-derivatives of it,...
12
votes
7answers
550 views

How can a set of components fail to make up a vector?

Many books in Physics insist to define vectors are objects with components with the property that the components transform in a proper way under a change of coordinates. Now, in mathematics, on the ...
11
votes
4answers
2k views

Is it strange that there are two directions which are perpendicular to both field and current, yet the Lorentz force only points along one of them?

By "strange" I mean 'Is there a reason for this, or is it something we accept as a peculiarity of our universe?' I see no reason why if magnetic field is in the $+x$ direction and a charge's velocity ...
11
votes
4answers
3k views

What would qualify as a deceleration rather than an acceleration if speed is unchanged?

The instantaneous acceleration $\textbf{a}(t)$ of a particle is defined as the rate of change of its instantaneous velocity $\textbf{v}(t)$: $$\textbf{a}(t)=\frac{\mathrm{d}}{\mathrm{d}t}\textbf{v}(t)....
11
votes
4answers
2k views

Is partial derivative a vector or dual vector?

The textbook(Introduction to the Classical Theory of Particles and Fields, by Boris Kosyakov) defines a hypersurface by $$F(x)~=~c,$$ where $F\in C^\infty[\mathbb M_4,\mathbb R]$. Differentiating ...
11
votes
8answers
2k views

Why do I get two results from a single free body diagram?

Calculating the component of the normal force (anti-)parallel to the gravitational force and setting them equal, I get \begin{equation} N \cos \theta = m\, g. \end{equation} On the other hand, ...
11
votes
4answers
40k views

Is there a general rule for determining the direction of tension force?

Tension, for me, is a tricky thing. After finishing a related chapter of my book and watching a video, I still can't get a hang of it. Here is a situation: My knowledge is that tension, just like ...
11
votes
4answers
848 views

Basis independence in Quantum Mechanics

The idea that the state of a system does not depend on the basis that we choose to represent it in, has always puzzled me. Physically I can imagine that the basis ought to just yield an equivalent ...
10
votes
6answers
3k views

Where am I confused about force addition?

As far as my knowledge is concerned, a vector quantity should possess magnitude and direction & more over it should also obey the laws of vector addition. As we all know that the vector sum of 3 ...
10
votes
6answers
728 views

Definition of inner product as in the case of work

According to the mathematical definition of "vectors", vectors are simply the elements of a set $V$ which forms a vector space structure $(V,F,+,*)$. The definition of inner product states that it is ...
10
votes
4answers
763 views

0-rank tensor vs vector in 1D

What is the difference between zero-rank tensor $x$ (scalar) and vector $[x]$ in 1D? As far as I understand tensor is anything which can be measured and different measures can be transformed into ...
10
votes
3answers
9k 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 him)...
10
votes
6answers
759 views

What does vector operator for angular momentum measure?

Consider the vector operator for angular momentum $\hat L=\hat L_x \vec i +\hat L_y \vec j + \hat L_z \vec k$. Does this mean that if we want to measure the angular momentum of a particle in state $\...
10
votes
3answers
6k views

Torque direction meaning

I apologize if this question is dumb, but I've looked all over for a straightforward answer and either I can't find one or the terms are too complex for me to understand. I have only a rudimentary ...
10
votes
1answer
568 views

Are the components of 4-vectors the physically measured quantities?

I am very confused with the difference between components of four-acceleration and coordinate acceleration. If I was in an inertial frame observing an accelerated object I would say its four-...
9
votes
3answers
1k views

Why do some people write the gravitational force as proportional to $\mathbf{r}/|\mathbf{r}|^3$?

I'm reading Mathematical Aspects of Classical and Celestial Mechanics, Second Edition by Arnold, Kozlov, and Neishtadt. It occurred to me that many people like to use third power when mention the law ...
9
votes
2answers
1k views

Infinite dimensional vector spaces vs. the dual space

I just happened across this over on Math Overflow. It references the following theorem from linear algebra: A vector space has the same dimension as its dual if and only if it is finite dimensional....
9
votes
2answers
2k views

Why is $\mathbf{B}$ a pseudovector?

I got the difference between polar vectors and axial vectors (pseudovectors). An example of pseudovector is $\mathbf{B}$. But why exactly the magnetic field is a pseudovector and its components ...
9
votes
1answer
4k 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 $$\pmb{H}=\pmb{\nabla} \phi +\pmb{\nabla} \times \pmb{G},$$ ...
9
votes
2answers
656 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 ...