What is Scalars: Definition and 67 Discussions

A scalar is an element of a field which is used to define a vector space. A quantity described by multiple scalars, such as having both direction and magnitude, is called a vector.In linear algebra, real numbers (or other elements of a field) are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a scalar to produce another vector. More generally, a vector space may be defined by using any field instead of real numbers, such as complex numbers. Then the scalars of that vector space will be the elements of the associated field.
A scalar product operation – not to be confused with scalar multiplication – may be defined on a vector space, allowing two vectors to be multiplied to produce a scalar. A vector space equipped with a scalar product is called an inner product space.
The real component of a quaternion is also called its scalar part.
The term is also sometimes used informally to mean a vector, matrix, tensor, or other, usually, "compound" value that is actually reduced to a single component. Thus, for example, the product of a 1 × n matrix and an n × 1 matrix, which is formally a 1 × 1 matrix, is often said to be a scalar.
The term scalar matrix is used to denote a matrix of the form kI where k is a scalar and I is the identity matrix.

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  1. P

    B Looking for help understanding scalars vs. vectors

    Why are pressure and energy not considered as vectors?
  2. R

    I Why is momentum considered a vector and kinetic energy a scalar?

    I'm not interested in the mathematical derivation, the mathematical derivation already is based on the assumption that momentum is a vector and kinetic energy is a scalar, thus it proves nothing. Specifically, what happens if we discuss scalarized momentum? What happens if we discuss vectorized...
  3. JackFyre

    B A doubt regarding vectors, scalars and their role in physics

    I have a doubt regarding the basic function of vectors and scalars in physics- What is the guarantee that every quantity(measured) in physics can be classified as either a vector or a scalar, and that while performing operations on said quantities, they will obey the already established rules...
  4. e2m2a

    I Confusion between vector components, basis vectors, and scalars

    There is an ambiguity for me about vector components and basis vectors. I think this is how to interpret it and clear it all up but I could be wrong. I understand a vector component is not a vector itself but a scalar. Yet, we break a vector into its "components" and then add them vectorially...
  5. H

    MHB Λ and μ are scalars, find the value of λ and the value of μ

    Figure shows a rectangle OABC in which OA = a and OC = c. F is the midpoint of CB and D is the point on AB such that AD : DB = 2:3 (a) Find _ _ _ _ _ __ (i) CF in terms of a _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (ii) AD in terms of c The lines OD and AF intersect at the point X Given that...
  6. karush

    MHB 307w.43.M40 find corresponding 3 scalars

    ok did an image due to notation to be correct, we will be talking about this in the zoom class next week but wanted to some grip of it before. Here is the RREFs $\text{rref}(B)=\left[ \begin{array}{cccc} 1 & 0 & 7 & 0 \\ 0 & 1 & -3 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right]$ I assume we can...
  7. George Keeling

    I Why are scalars and dual vectors 0- and 1-forms?

    I am told: "A differential p-form is a completely antisymmetric (0,p) tensor. Thus scalars are automatically 0-forms and dual vectors (one downstairs index) are one-forms." Since an antisymmetric tensor is one where if one swaps any pair of indices the value of the component changes sign and 1)...
  8. M

    A Why are open strings vectors or scalars, or massive?

    In string theory, if we have NN BCs along ##X^i, i = 1, \ldots, n-1## and DD BCs along ##X^a, a = n, \ldots, 25## then you get, from ##\alpha^{i,a}_{-1}|0,p\rangle ##, ##n## massless vectors and ##24-n## massless scalars. I understand that for the first excited level, ##M^2=0## and so we have...
  9. F

    B Explaining vector & scalar quantities to a layman

    I've been asked by someone with minimal background in physics to explain what vector and scalar quantities are and give examples. Here are my thoughts: A scalar is a quantity that has a magnitude only, it is completely specified by a single number. Importantly, it has no directional dependence...
  10. F

    I Vector components, scalars & coordinate independence

    This question really pertains to motivating why vectors have components whereas scalar functions do not, and why the components of a given vector transform under a coordinate transformation/ change of basis, while scalar functions transform trivially (i.e. ##\phi'(x')=\phi(x)##). In my more...
  11. Safinaz

    I LHC searchs for charged Higgs scalars

    Hi all, I'd like to know what is the most recent exclusion bounds on the mass of new charged Higgs scalar according to CMS and ATLAS collaborations searches at CERN ..
  12. S

    B Product of scalars vs vectors

    (Scalar)·(Scalar) = Scalar (Scalar)·(Vector) = Scalar (Vector)·(Vector) = Scalar (Scalar)x(Scalar) = Not valid (Scalar)x(Vector) = Vector (Vector)x(Vector) = VectorDid I get them right, if not why? Thanks
  13. S

    A Calculate the Weyl and Ricci scalars for a given metric

    hello dear, I need to calculate the Weyl and Ricci scalars for a given metric. let's assume for a kerr metric. by using the grtensor-II package in maple i am not able to get the results. would anyone help me out. for ordinary metric like schwarzschild metric, However its not working for Kerr...
  14. Math Amateur

    MHB Extension of Scalars: Dummit & Foote's Section 10.4 Q&A

    I am reading Dummit and Foote's book: Abstract Algebra ... ... and am currently focused on Section 10.4 Tensor Products of Modules ... ... I have a basic question regarding the extension of the scalars ... Dummit and Foote's exposition regarding extension of the scalars reads as...
  15. Math Amateur

    I Extension of scalars .... D&F, Section 10.4: Tensor Products

    I am reading Dummit and Foote's book: Abstract Algebra ... ... and am currently focused on Section 10.4 Tensor Products of Modules ... ... I have a basic question regarding the extension of the scalars ... Dummit and Foote's (D&Fs) exposition regarding extension of the scalars reads as...
  16. bcrowell

    Eigenvalues of curvature tensors as curvature scalars?

    I've been playing around with the Carminati-McLenaghan invariants https://en.wikipedia.org/wiki/Carminati–McLenaghan_invariants , which are a set of curvature scalars based on the Riemann tensor (not depending on its derivatives). In general, we want curvature scalars to be scalars that are...
  17. E

    Why Complex Scalars in 4D Supersymmetric Theories?

    The scalar fields of supersymmetric theories in 4 spacetime dimensions are a set of complex fields (usually denoted by ##z^{\alpha}##). How can this be physically translated? More precisely, we know that in 5D, those scalars are real, so what is that makes them real here but complex there?
  18. E

    Scalars and special geometry

    Crossing over the following paragraph: There are three types of special manifolds which we shall discuss, related to the real scalars of gauge multiplets in D = 5, the complex scalars of D = 4 gauge multiplets and the quaternionic scalars of hypermultiplets. Since there are no scalars in the...
  19. Mr Davis 97

    Difference between scalars and one-dimensional vectors?

    What is the difference between scalars and one-dimensional vectors? I know that we represent the set of all two-dimensional vectors as ##\mathbb{R}^2##, so doesn't this mean that we would represent the set of all one-dimensional vectors as as just ##\mathbb{R}##? However, doesn't this also refer...
  20. D

    Lorentz scalars - transformation of a scalar field

    When one considers a Lorentz transformation between two frames ##S## and ##S'##, such that the coordinates in ##S## are given by ##x^{\mu}## and the coordinates in ##S'## are given by ##x'^{\mu}##, with the two related by x'^{\mu}=\Lambda^{\mu}_{\;\;\nu}x^{\nu} then a scalar field ##\phi (x)##...
  21. T

    Expanding a function in terms of a vector

    Homework Statement ## L (v^2 + 2 \pmb{v} \cdot \pmb{ \epsilon } ~ + \pmb{ \epsilon} ^2)##, where ## \pmb{\epsilon}## is infinitesimal and ##\pmb{v}## is a constant vector (## v^2 ## here means ## \pmb{v} \cdot \pmb{v} ## ), must be expanded in terms of powers of ## \pmb{\epsilon} ## to give...
  22. Safinaz

    Color factors of color -- octet scalars

    Hi all, Has anyone an idea how can we derive the color factor ## C_2## , eq. 27 in http://authors.library.caltech.edu/8947/1/GREprd07.pdf ? ## C_2 ## belongs to the second Feynman diagram in fig. 3 which includes two gluons and 4 octet scalars . Best.
  23. P

    Answer Scalar Quantity Q: Is Everything False?

    <<Moderator note: Missing template due to originally being posted in different forum.>> Kindly throw me light on this! Q: A scalar quantity is one that: a) is conserved in a process. b) can never take negative values c) Must be dimensionless. d) Does not vary from one point to another in...
  24. Safinaz

    Couplings of new Higgs scalars and CP- invariance

    Hi all, In A. Djouadi's review for Higgs, volume II, " arXiv:hep-ph/0503173v2 ", Sec. 1.2.3, it discuss the couplings of SUSY new scalars with gauge bosons, there are some points I don't understand: - CP–invariance forbids WWA, ZZA and W ZH ± couplings - For the couplings between two Higgs...
  25. caffeinemachine

    MHB Understanding Extension of Scalars in a Vector Space

    $\newcommand{\R}{\mathbf R}\newcommand{\C}{\mathbf C}$ Low-Tech Complexification: Let $V$ be a finite dimensional vector space over $\R$. We can forcefully make $W:=V\times V$ into a complex vector space by defining addition component-wise and product $\C\times W\to W$ as $$ (a+ib)(u...
  26. J

    Identifying Scalars and Vectors: A Guide

    hai, can anybody say how to identify scalar and vectars
  27. Adoniram

    Sakurai - Constructing 2x2 matrix from scalars?

    Hello all, I am beginning a course in QM with Sakurai's 2nd Edition book on QM. In one of our problems, he defines a matrix as the sum of a scalar and a dot product... This seems like nonsense to me, but he uses the same notation in the next problem, so I am guessing this is some unorthodox...
  28. Safinaz

    Constrains on new Higgs scalars from LHC recent data

    Hi all, I wonder if I study new Higgs scalars, How the data of the LHC for searching for heavier scalars in h-> WW->lvlv and h -> ZZ-> 4 l channels like in [arXiv:1304.0213] can make constrain on my study for the new Higgs? How a figure like figure 2 can give me data for my model...
  29. N

    How many scalars are needed in GUT theories?

    Hello, I don't know much about GUT physics, but I've been wondering whether these models usually breaks the grand unified symmetry to the standard model all at once, or multiple times at different energies. And in the case of multiple breakings, how many Higgs-like scalars are needed...
  30. A

    Scalars as Rank 0 Tensors?

    Homework Statement I'm learning a bit about tensors on my own. I've been given a definition of a tensor as an object which transforms upon a change of coordinates in one of two ways (contravariantly or covariantly) with the usual partial derivatives of the new and old coordinates. (I...
  31. C

    Linear Algebra, scalars to find det(A)

    Homework Statement Let X1, X2,...,Xn be scalars. Calculate det(A) where A= nxn matrix with [ x1+1 x1+2...x1+n x2+1 x2+2...x2+n ... ... ... xn+1 xn+2...xn+n] Homework Equations det(A) =...
  32. Math Amateur

    MHB Tensor Products - D&F - Extension of the scalars

    I am attempting to understand Dummit and Foote exposition on 'extending the scalars' in Section 10.4 Tensor Products of scalars - see attachment - particularly page 360) [I apologise in advance to MHB members if my analysis and questions are not clear - I am struggling with tensor products! -...
  33. M

    Scalars and determining subspaces

    To determine if a subset of a vector space is a subspace, it must be closed under addition and scalar multiplication. As far as I can tell, this means adding two arbitrary vectors in the subset and having the sum be within the subset. But...can the scalar be any number? Is there any limitation?
  34. D

    Linear Transformation from R^m to R^n: Mapping Scalars to Vectors

    Can we think of a linear transformation from R^m-->R^n as mapping scalars to vectors? Let me say what I mean. Say we have some linear transformation L from R^m to R^n which can be represented by a matrix as follows: L=[ a11x1+a12x2+...+a1mx m a21x1+... . . . anmx1+...+ anmxm...
  35. N

    Breaking Symmetries: 20 Real Scalars and 24 Symmetries

    Homework Statement Suppose that there is a gauge group with 24 indepenent symmetries and we find a set of 20 real scalar fields such that the scalar potential has minima that are invariant under only 8 of these symmetries. Using the Brout-Englert-Higss mechanism, how many physical fields are...
  36. B

    Vector & Scalar Homework: Base Camp to Lake B

    Homework Statement A plane flies from base camp to Lake A, 280 km away inthe direction 20.0°north of east. After dropping off sup-plies, it flies to Lake B, which is 190 km at 30.0°west of north from Lake A. Graphically determine the distance and direction from Lake B to the base camp...
  37. A

    Fermions in Early Universe: Why We Ignore Them?

    When we talk about the inflation and other cosmological topics, we calculate the scalar dynamics in the early universe. But how do fermions behave? In principle they should carry same amount of energy and they can effect the evolution of scalars via interactions. Why we just ignore them? Is...
  38. N

    Vectors and scalars question

    Question: Solve the following vector equations mathematically: a) a + b b) c- d c) 2a -1/2b d) a + c +b Given: a = 10m[R] b = 20m[L] c = 10m[U] d = 5m[D] Solution: a) 10m [R] - 20m [L] = 10m [L] b) 10m [U] - 5m [D] = 5m [U] c) 20m [R] - 10m [L] = 10m [R] d) a + c +b = 10m[R] + 10m...
  39. B

    Solving Matrix Equations with Real Scalars

    Hello, A(rB) = r(AB) =(rA)B where r is a real scalar and A and B are appropriately sized matrices. How to even start? A(rbij)=A(rB), but then you can't reassociate... Also, a formal proof for Tr(AT)=Tr(A)? It doesn't seem like enough to say the diagonal entries are unaffected by...
  40. T

    Quantum states as scalars

    Title may sound weird,but I think it might be worth exploring In axiomatic formulation of quantum mechanics, quantum states are postulated as vectors residing in Hilbert space. The only apriori requirement that Iam aware of ,for a quantity to qualify as a quantum state, is that it should...
  41. M

    What does the R-Symmetry do to scalars and fermions in N=4 U(N) SYM?

    Hi, I have forgotten all about my supersymmetry knowledge and all about my group theory knowledge. I am trying to understand what the R-symmetry in N=4 U(N) SYM does. Sadly I have never actually learned anything about supersymmetry which is larger than N=1. I know the R-symmetry is SU(4) and...
  42. E

    Real scalars for complex scalar results

    Hi guys, I'm a bit puzzled. I'm just reading some offline lecture notes where the Feynman rules of real (!) scalars coupled to gluons are given. However, with these rules the amplitude for phi g -> \bar{phi} g is considered. There are no further instructions. I'm just wondering how one can...
  43. M

    Scalars and Vectors: The Difference

    Hey. :) I have just come onto working with vectors in pure mathematics and have no problems calculating with them. However, I do not really understand the difference between a vector and a scalar. A scalar has magnitude only. A vector has magnitude and direction. Since trying to...
  44. T

    Scalars vs Vectors: Angular Displacement, Speed, Flux, Potential & More

    Homework Statement Angular displacement, angular speed, , magnetic flux, electric potential, E.M.F., P.D., gravitational potential, stress,inductance. Which are the scalars and vectors? Homework Equations Scalars are quantities, which have magnitude only, whereas vectors have both magnitude and...
  45. J

    Divergence of curvature scalars * metric

    How can one work out what terms like: (g^{cd}R^{ab}R_{ab})_{;d} are in terms of the divergence of the Ricci curvature or Ricci scalar? One student noted that since: G^{ab} = R^{ab} - \frac12 g^{ab}R {G^{ab}}_{;b} = 0 that we could maybe use the fact that G^{ab}G_{ab} = R^{ab}R_{ab} - \frac12...
  46. D

    Scalars z such that A-zI is singular

    Homework Statement Consider the (2x2) symmetric matrix A = [a b; b d] Show that there are always real scalars z such that A-zI is singular [Hint: Use quadratic formula for the roots from the previous exercise ] t^2 - (a + d)t + (ad - bc) =0 Homework Equations The Attempt at a...
  47. Char. Limit

    Vector Math: Squares, Roots, Logarithms

    If someone takes a vector, and squares it, does it become a scalar? Also, is it possible to take the square root of a vector, and would the result be a vector or a scalar? Lastly, the logarithm of a measurement is dimensionless. However, if you raise the base of that logarithm to the power...
  48. J

    Help Needed Solving For Two Scalars Such That au + bw = v

    Homework Statement Find a and b such that v= au + bw given that v = <1, 1> Homework Equations v = au + bw, v = <1,1> The Attempt at a Solution I have tried to solve for the appropriate values via guess and check, but it hasn't worked out. I cannot find a proportion either...
  49. R

    Projection of vectors and scalars

    Hi I was wondering if someone can explain what projection of vectors and scalars mean. I read a lot of site but they fail to give me a clear explanation. Thanks.
  50. C

    Scalars, vectors, pseudo-scalars, pseudo-vectors

    This is a basic question about the scalars, vectors, pseudo-scalars, and pseudo-vectors. I know that scalars and pseudo-vectors don't change sign under parity and vectors and pseudo-scalars do, but does that imply that scalars have to be even function of x, y, z (like for example x^2+y^4+z^2)...
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