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

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  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

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  6. karush

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  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?

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  9. F

    B Explaining vector & scalar quantities to a layman

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  10. F

    I Vector components, scalars & coordinate independence

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  11. Safinaz

    I LHC searchs for charged Higgs scalars

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  12. S

    B Understanding Scalar and Vector Products in Geometric Algebra

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  13. S

    A Calculate the Weyl and Ricci scalars for a given metric

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  14. Math Amateur

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

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  15. Math Amateur

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

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  16. bcrowell

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  17. E

    Why Complex Scalars in 4D Supersymmetric Theories?

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  18. E

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  19. Mr Davis 97

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  20. D

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  21. T

    Expanding a function in terms of a vector

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  22. Safinaz

    Color factors of color -- octet scalars

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  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

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  25. caffeinemachine

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  26. J

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  27. Adoniram

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  28. Safinaz

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  29. N

    How many scalars are needed in GUT theories?

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  30. A

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  31. C

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  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

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  34. D

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

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  35. N

    Breaking Symmetries: 20 Real Scalars and 24 Symmetries

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  36. B

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  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

    How Do You Solve Vector Equations with Given Values and Unknowns?

    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

    Can Scalars Represent Quantum States Effectively?

    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

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  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

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  49. R

    Projection of vectors and scalars

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  50. C

    Scalars, vectors, pseudo-scalars, pseudo-vectors

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