scalar

(noun)

A quantity that has magnitude but not direction; compare vector.

Related Terms

  • magnitude
  • unit vector
  • superposition
  • vector

(noun)

A quantity which can be described by a single number, as opposed to a vector which requires a direction and a number.

Related Terms

  • magnitude
  • unit vector
  • superposition
  • vector

Examples of scalar in the following topics:

  • Multiplying Vectors by a Scalar

    • While adding a scalar to a vector is impossible because of their different dimensions in space, it is possible to multiply a vector by a scalar.
    • A scalar, however, cannot be multiplied by a vector.
    • To multiply a vector by a scalar, simply multiply the similar components, that is, the vector's magnitude by the scalar's magnitude.
    • Multiplying vectors by scalars is very useful in physics.
    • For example, the unit of meters per second used in velocity, which is a vector, is made up of two scalars, which are magnitudes: the scalar of length in meters and the scalar of time in seconds.
  • Unit Vectors and Multiplication by a Scalar

    • Multiplying a vector by a scalar is the same as multiplying its magnitude by a number.
    • In addition to adding vectors, vectors can also be multiplied by constants known as scalars.
    • Examples of scalars include an object's mass, height, or volume.
    • When multiplying a vector by a scalar, the direction of the vector is unchanged and the magnitude is multiplied by the magnitude of the scalar .
    • (iii) Increasing the mass (scalar) increases the force (vector).
  • Scalars vs. Vectors

    • Physical quantities can usually be placed into two categories, vectors and scalars.
    • In contrast, scalars require only the magnitude.
    • Scalars differ from vectors in that they do not have a direction.
    • Scalars are used primarily to represent physical quantities for which a direction does not make sense.
    • This video introduces the difference between scalars and vectors.
  • Introduction to Scalars and Vectors

    • Given this information, is speed a scalar or a vector quantity?
    • Speed is a scalar quantity.
    • Distance is an example of a scalar quantity.
    • Scalars are never represented by arrows.
    • (A comparison of scalars vs. vectors is shown in . )
  • Matrix and Vector Norms

    • For scalars, the obvious answer is the absolute value.
    • The absolute value of a scalar has the property that it is never negative and it is zero if and only if the scalar itself is zero.
    • A norm is a function from the space of vectors onto the scalars, denoted by $\| \cdot \|$ satisfying the following properties for any two vectors $v$ and $u$ and any scalar $\alpha$ :
  • Linear Vector Spaces

    • The definition of such a space actually requires two sets of objects: a set of vectors $V$ and a one of scalars $F$ .
    • For our purposes the scalars will always be either the real numbers $\mathbf{R}$ or the complex numbers $\mathbf{C}$ .
    • Addition and scalar multiplication are defined component-wise:
    • This implies the uniqueness of the zero element and also that $\alpha \cdot 0 = 0$ for all scalars $\alpha$ .
    • And the minus element is inherited from the scalars: $[-f](t) = -f(t)$ .
  • Components of a Vector

    • This differentiates them from scalars, which are mere numbers without a direction.
    • Andersen explains the differences between scalar and vectors quantities.
  • Superposition of Electric Potential

    • Recall that the electric potential V is a scalar and has no direction, whereas the electric field E is a vector.
    • This is consistent with the fact that V is closely associated with energy, a scalar, whereas E is closely associated with force, a vector.
    • Summing voltages rather than summing the electric simplifies calculations significantly, since addition of potential scalar fields is much easier than addition of the electric vector fields.
    • The electric potential at point L is the sum of voltages from each point charge (scalars).
  • Tensors

    • We have essentially stumbled upon a few nice four-vectors, but there is a more systematic way of dealing with four-vectors, scalars and other quantities like the transformation matrix $\Lambda^\mu_{~\nu}$.
    • Let's say there is a scalar field defined over all spacetime.
    • The quantity on the left is clearly a scalar because it is the different in the value of a scalar field at two points.
    • If we take $A^{\mu}$ to be the vector potential plus the scalar potential,
  • Phase-Space Density

    • that transforms as a scalar where $n(x^\alpha)$ is the number density.
    • One could use it as the source for a scalar theory of gravity, but it would violate the equivalence principle.
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