Scalar

Introduction 

Two types of mathematical values represent object movement in today’s world. The terms force, speed, velocity, and work are frequently used, and each of these quantities are classified as  either scalar or vector quantities. Scalar quantities, such as mass and electric charge, are physical quantities that have only magnitudes.

Scalar Definition

Scalar quantities are defined as physical quantities that simply have magnitude and no direction of motion. Some physical quantities can be characterised solely by their numerical value, without the need for additional information. It is only the magnitudes of these physical values that are added in this case, as the addition of these quantities follows the specific laws of algebra.

The Physical Quantity of Scalar

A physical amount of scalar, like other physical quantities, is often stated by a numerical value and a physical unit, rather than just a number, to provide its physical meaning. It can be thought of as the result of multiplying the number by the unit. The length of each base vector in a coordinate system where the base vector length matches the physical distance unit in use has no bearing on the physical distance. A physical distance differs from a metric in that it is not just a real number, although the metric is calculated to a real number. However, the metric may be converted to the physical distance by converting each base vector length to the physical unit. 

The formula for computing scalars may be affected by changes in the coordinate system, but not the scalars themselves. A change in the coordinate system does not affect vectors, but it does modify their descriptions. 

Non-Relativistic Scalars 

Temperature

Temperature is a scalar quantity. A single integer represents the temperature at a given location. On the other hand, velocity is a vector quantity.

Other examples

Mass, charge, volume, time, speed, and electric potential at a point inside a medium are all examples of scalar quantities in physics. 

In the three-dimensional space, the distance between two points is a scalar. Still, the direction between them is not because expressing a direction necessitates the use of two physical quantities: the angle on the horizontal plane and the angle away from it as force has both direction and magnitude, it cannot be stated using a scalar; however, the magnitude of a force may be described using a scalar. For example, the magnitude of a particle’s gravitational force is not a scalar, but the force itself is. The velocity of an item is not a scalar, but its speed is. 

Electric charge and charge density are two more scalar concepts in Newtonian mechanics.

Relativistic scalars

Changes in coordinate systems that trade space for time are considered in the theory of relativity. As a result, in ‘classical’ (non-relativistic) physics, some physical quantities that are scalars must be joined with other numbers and regarded as four-vectors or tensors. To make a relativistic 4-vector, the charge density at a location in a medium, which is a scalar in classical physics, must be joined with the local current density (a 3-vector). Energy density must be paired with momentum density and pressure in the stress-energy tensor.

Electric charge, spacetime interval (for example, proper time and proper length), and invariant mass are examples of scalar quantities in relativity.

Vector Quantities

Vector quantities are physical quantities that are characterised by the presence of both magnitude and direction in their magnitude and direction. For instance, displacement, force, torque, momentum, acceleration, velocity, etc.

Scalar Product of Two Vectors

The product of two vectors yields either a number or a scalar. Scalar products are advantageous for defining energy and work relationships. The scalar product of force and displacement vectors represents the work done by a force (which is a vector) in displacing an object (which is also a vector). The scalar product is symbolised by a dot ( . ), and its formula is as follows:

Xˆ . Yˆ = XY Cos ፀ, where ፀ is the angle between the vectors.

Due to the dot notation employed in it, the scalar product is sometimes known as the dot product.

Characteristics of the Scalar Product of Two Vectors

• Commutative is the scalar product.
• A scalar product’s two mutually perpendicular vectors are equal to zero.
• The product of the magnitudes of the two parallel and vector components of a scalar product is equal to the product of their magnitudes.
• The square of its magnitude equals the vector’s self-product.

Scalar Product Formula

The scalar product of a and b is: a · b = |a| |b| cos θ 

To remember this formula, we can think of it as Modus of the first vector, multiplied by the Modus of the second vector, multiplied by the Angle of the first and second vectors.

Applications of the Scalar Product

• Scalar products are useful in geometry because they may be used to detect the direction between arbitrary vectors in space.
• With the scalar product, it is possible to calculate the cosine of an angle by using the components of two vectors, and the magnitudes, A and B, can be obtained from the components.
• The scalar product can also be used to express magnetic potential energy and the potential of an electric dipole, amongst other applications.

Applications of Vector

In the physical sciences, vectors are critical. Vectors can be used to represent any quantity that has magnitude and direction and satisfies the vector addition requirements. 

One such term is velocity, which relates to the magnitude of a speed. For instance, the vector (0, 5) (in two dimensions, with the positive y-axis labelled ‘up’) could represent an upward velocity of 5 metres per second. Force is another quantity that may be represented by a vector because it has a magnitude and a direction and satisfies the vector addition principles. 

Many other physical variables are described using vectors, including linear displacement, displacement, linear acceleration, angular acceleration, linear momentum, and angular momentum. Other physical vectors, such as electric and magnetic fields, are represented by vector fields, which are composed of a collection of vectors at each point in space.

Conclusion 

Scalar quantities are those which can only measure the quantity, not the direction. So, in the calculation of scalar quantity, we only have to deal with quantities, and these quantities can be obtained by Newton’s law applications of vectors. As it only deals with the magnitude, mostly quantities like mass, volume, time, temperature, energy, distance, which we use in daily life are scalar quantities. Although, scalar quantity can also be defined as a size of any quantity.