Why are dot products scalar (in the general sense) if seen from an intuitive point of view ? i am not asking about a particular case like kinetic energy or work but a i want the answer for general dot products
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I get the physical significance of vector addition &amp; subtraction. But I don't understand what do dot &amp; cross products mean?   More specifically,    Why is it that dot product of vectors $\vec{A}$ and $\vec{B}$ is defined as $AB\cos\theta$? Why is it that cross product of vectors $\vec{A}$ and $\vec{B}$ is defined as $AB\sin\theta$, times a unit vector determined from the right-hand rule?   To me, both these formulae seem to be arbitrarily defined (although, I know that it definitely wouldn't be the case).   If the cross product could be defined arbitrarily, why can't we define division of vectors? What's wrong with that? Why can't vectors be divided?
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Every now and then a question comes up about the status of the momentum operator in the infinite square well, and while we have two good answers on the topic  and , I'm generally not satisfied by their level of detail and by how easy (not very) it is to find them when you need them. I therefore posit that it is high time we settle things and build a canonical Q&amp;A thread for this, so, with that in mind:  What's the deal with momentum in the infinite square well?   Is the momentum operator $\hat p=-i\hbar \frac{\mathrm d}{\mathrm dx}$  when restricted to the compact interval of the well? Are there any subtleties in its definition, via its domain or similar, that are not present for the real-line version? Is the momentum operator $\hat p=-i\hbar \frac{\mathrm d}{\mathrm dx}$  in these conditions? If not, why not, and what are the consequences in terms of the things we normally care about when doing one-dimensional QM? If it is not self-adjoint, does it admit a ? If it does, is that extension unique? If the extension is not unique, what are the different possible choices, and what are their differences? Do those differences carry physical meaning / associations / consequences? And what is a self-adjoint extension anyways, and where can I read up about them? What is the spectrum and eigenvectors of the momentum operator and its extensions? How do they differ from each other? Is there such a thing as a momentum representation in this setting? If not, why not? What is the relationship between the momentum operator (and its possible extensions) and the hamiltonian? Do they commute? Do they share a basis? If not, why not? Do these problems have counterparts or explanations in classical mechanics? ()   And, more importantly: what are good, complete and readable references where one can go and get more information about this?
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Why is the Lagrangian a function of the position and velocity (possibly also of time) and why are dependences on higher order derivatives (acceleration, jerk,...) excluded?  Is there a good reason for this or is it simply "because it works".
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I read  a while ago, and while thinking about $\nabla$, I realized something. Since the cross product can be written as a determinant, in higher dimensions we require extra vector inputs. IIRC it's called the "wedge product" in higher dimensions.  Alright, how does this work when we generalize Maxwell to higher dimensions? Curl can be written (abuse of notation, yes) as a cross product with $\nabla$. But, to generalise it to higher dimensions, we need multiple inputs. We need something like $\nabla_4(\mathbf{B_1},\mathbf{B_2})$  in four dimensions, and so on. So we have two ways to get out of this: We can either use a different way to write curl in multiple dimensions (The  has stuff on this which I don't understand), or there are more than one $E$ and $B$ fields in higher dimensions.  So which is it? How are Maxwell's laws generalised to higher dimensions?  Just a note: I never understood the linked answer after the first sentence (didn't know enough), so if there's something obvious there that answers this question, I missed it. I know nothing of higher-dimensional analysis, so if complex notation is going to be unavoidable for the multiple dimensions, I'd be fine if you showed me what happens in four dimensions.
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Various textbooks that I am currently consulting (including Spacecraft Dynamics and Control An Introduction - Anton H.J. De Ruiter | Christopher J. Damaren | James R. Forbes Section 1.4, page 32) use $\delta$, not $d$ or $\partial$ to express an infinitesimal quantity. In the context of the reference text above, the symbol is used as part of a general definition for the derivative of a vector. Specifically, given   $$\mathbf{\vec{r}}=\mathcal{\vec{F^{T}_{1}}}\mathbf{\vec{r_{1}}}$$   where   $$\mathcal{\vec{F^{T}_{1}}}$$   is a vectrix   $$[\mathbf{\vec{x_1}} \phantom{s} \mathbf{\vec{y_1}} \phantom{s} \mathbf{\vec{z_1}}].$$  The time derivative of the vector is defined as   $$\dot{\mathbf{\vec{r}}}\triangleq \lim_{\delta t\to 0}\frac{\delta\mathbf{\vec{r}}}{\delta t}.$$  In this context, what is the difference between $\frac{\delta\mathbf{\vec{r}}}{\delta t}$ and $\frac{d \mathbf{\vec{r}}}{d t}$ ?  Note about thermodynamic use of $\delta$: My understanding is that $\delta$ is used in thermodynamic equations to express path dependence of a scalar quantity such as heat or work. What does it mean in the context of an abstract physical vector?
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from a while ago and answers/comments to  from earlier today both make heavy mention of the fact that fluid pressure is a scalar.  Although this information was surprising to me, it makes sense intuitively, especially when considering Pascal's Principle and liquid depth pressure.  However, I struggle when thinking about the mathematical reasons why pressure is a scalar, and the answers to the first question linked either appealed to intuition, seemed contentious, or unsatisfying.  My confusion comes from the fact that pressure is often defined at the high school/undergrad level as $$P\equiv \frac{F}{A}$$ where this not just a relationship, but often presented as the equation definition of pressure.  Force is a vector, and I had assumed that area was a scalar.  Therefore, I had always figured this was a scalar multiplication problem, and the product would also be a vector.  Now that I know that pressure is a scalar, I just can't see how you would get a scalar for pressure out of this relationship.  I see two possibilities:  1) For reasons I don't currently understand, area is treated as a vector in continuum physics.  As a result, some type of vector algebra must be applied to divide a vector by another vector in order to yield a scalar.  If this is the case, I would like to see at least a framework for this math in the answer.  2) If possibility 1 is not the case, then the other possibility in my mind is that $P=\frac{F}{A}$ is not truly the definition of pressure, and there exists some more basic definition for pressure that would explain its scalar nature.  Are either of these explanations correct, or is there a third possibility that I have not considered?  I've tried to make this a substantially different question than either of the other two questions that I've linked to, but if that does not appear to be the case, please let me know.
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I'm searching to develop the intuition (rather than memorization) in relating the two forms of a dot product (by an angle theta between the vectors and by the components of the vector ).      For example, suppose I have vector $\mathbf{a} = (a_1,a_2)$ and vector $\mathbf{b}=(b_1,b_2)$.  What's the physical or geometrical meaning that   $$a_1b_1 + a_2b_2 = |\mathbf{a}||\mathbf{b}|\cos(\theta)\;?$$  Why is multiplying $|\mathbf{b}|$ times $|\mathbf{a}|$ in direction of $\mathbf{b}$ the same as multiplying the first and second components of $\mathbf{a}$ and $\mathbf{b}$ and summing ?       I know this relationship comes out when we use the law of cosines to prove, but even then i cant get a intuition in this relationship.  This image clarifies my doubt:    Thanks
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Most questions usually just relate to what these can be used for, that's fairly obvious to me since I've been programming 3D games/simulations for a while, but I've never really understood the inner workings of them... I could get the cross product equation as a determinant of a carefully-constructed matrix,   but what I want to ask is... How did the dot and cross product come to be? When were they "invented"? Some detailed proofs? Did someone say: "Hey, wouldn't it be nice if we could construct a way to calculate a vector that is perpendicular to two given operands?"  Basically, how/why do they work?  I would appreciate explanations, links to other explanations, other web resources... I've been searching the Internet lately for explanations, but most of them are on how to use it and nothing that really gives substance to it.
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As far as I've seen, there are only two types of vector multiplication defined, dot product/inner product which is defined in a vector space of any dimension, even infinite, and cross product, which is only defined in three dimensions. Are there more definitions of vector products?
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Can anyone think of an intuitive explanation of the product rule? I'm not asking for a derivation. To me it seems like quite an un-untuitive result, as apposed to the chain-rule (which is ironically harder to derive).
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Why the log? Is it there to make the growth of the function slower?  As this is a common experimental observable, it doesn't seem reasonable to take the range from $[0,\infty)$ to $(-\infty,\infty)$ (For a particle emitted along the beam axis after collision $\theta = 0$ wouldn't be better to have a number that says how close it is to zero rather than one that says how large a number it is. I hope that makes the question clear.)
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I know how to CALCULATE dot product, but still don't understand MEANING of it.  I mean, If I have to explain vector addition I would say that "Sum of two vectors give you the third vector that has the same effect as other two combined"  But I don't know how to explain dot product. For example we have two vectors: v=(3,1) and w=(2,4). Dot product of these two is 3*2 + 1*4 = 10. What does this result (10) tell us?
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I have a question about an argument used in Schwabl's "Advanced Quantum Mechanics" concerning the properties of the Klein-Gordan-Equation (see page 120):    Since the eigenenergies of free solutions are $E= \pm \sqrt{p^2c^2+m^2c^4}$the energy states aren't bounded from below. But I don't understand why that  then K-G-equation provide a scalar theory that does not contain spin and then could only describe particles with zero spin.   Intuitively, I guess because that spin can't regard by a scalar equation but I find this "argument" too squishy and would like to know a more plausible argument.
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I noticed something when I was doing a proof of the BAC-CAB rule, and wanted to check if my intuition was correct.   First, when I actually multiplied out $\vec{B} (\vec{A} \cdot\vec{C})-\vec{C}(\vec{A}\cdot\vec{B}) $ I noticed that I got the same thing. That is, the two quantities are equal. This makes sense since they are dot products. But that further implies that the expression goes to zero.  Well, that means that $\vec{A} \times (\vec{B} \times \vec{C})$ would have to be zero too. And when I thought about it it occurred to me that $\vec{B} \times \vec{C}$ describes a vector perpendicular to both B and C; it's orthogonal. But $\vec{A} \times$ (any vector) will describe a vector perpendicular to (any vector) and in this case, that's $\vec{B} \times \vec{C}$. That is going to be parallel with the vector described by $\vec{B} \times \vec{C}$ itself and that further implies that the cross product of the two is equal to zero. Which is equal to the BAC-CAB quantity above.   So, what is the big honking flaw in my reasoning here? :-)
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I understand that a vector has direction and magnitude whereas a point doesn't.  However, in the course notes that I am using, it is stated that a point is the same as a vector.  Also, can you do cross product and dot product using two points instead of two vectors? I don't think so, but my roommate insists yes, and I'm kind of confused now.
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In  it's said that time is a scalar quantity. But its hard to understand that how? As stated that we consider only the magnitude of time then its a scalar. But on basis of time we define yesterday, today and tomorrow then what it will be?
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The action  $$S=\int L \;\mathrm{d}t$$  is an important physical quantity. But can it be understood more intuitively? The Hamiltonian corresponds to the energy, whereas the action has dimension of energy × time, the same as angular momentum.  I've heard the action being described as a measure of change, although I don't know how this description can be justified.