Fundamentally, physics is the mathematical representation of nature. Almost every principle and concept has a mathematical background. There are countless instances where mathematical techniques are essential to dissect and explain the concepts. The mathematical aspect of the physical concept is the basis of physics. With the help of mathematics, any complex concept can be turned into understandable sets of laws, formulas, or equations. The combination of necessary words can describe the motion of bodies. But words cannot explain the complex physics behind motion, and only mathematical methods can decipher and explain these complex concepts consistently. Various mathematical quantities are used to describe physical phenomena. Speed, velocity, acceleration and displacement are described using mathematical quantities.
Mathematical quantities used to represent physical phenomena are divided into two distinct categories: scalars and vectors. A scalar is a quantity that has only magnitude, whereas a vector has magnitude and a distinct direction.
The video explains the basics of vector algebra and 3D geometry
Scalar Quantity
In physical science, scalars are physical quantities that only have magnitude. The direction or orientation doesn’t matter in their functionalities, and the magnitude completely denotes them. Some familiar scalar examples are density, energy, mass, time, volume and speed. Complex quantities like force, velocity and acceleration have direction and magnitude (vectors). Scalars are denoted by real numbers that are generally positive and rarely negative. The work done on a body by force is a negative quantity when the body travels in the opposite direction to that the force exerts. Scalars can be calculated using the basic laws of algebra.
Vector Quantity
In mathematics and physics, a vector is a geometric quantity with magnitude and direction, and it is also called a spatial vector or Euclidean vector or geometric vector. A vector can be added using vector algebra. A Euclidean vector is often denoted by a directed line shape or pictorially as an arrow joining a beginning point A with an ending point B.
A vector is what is required to displace point A to point B. The vector means “carrier” in Latin. Astronomers originally used the term vector around the 18th century to research planetary movement around the Sun. A vector’s magnitude is the distance between two points. The direction of the vector refers to the direction of the movement from A to B. Various algebraic methods of real numbers, such as multiplication, negation, subtraction and addition, have similar analogues for vectors. They are compatible with conventional algebraic laws of associative, distributive and commutative.
Vectors play a crucial role in physics. The acceleration and velocity of a body and its forces can all be represented by vectors. Numerous other physical quantities can be considered as vectors. Even though most of them do not denote distance, their directions and magnitude can still be denoted by the direction of an arrow and length. The mathematical description of physical vectors depends on the coordinate system employed to describe them.
The main examples of vector quantity are:
- Linear momentum
- Acceleration
- Displacement
- Momentum
- Angular velocity
- Force
- Electric field
- Polarisation
Important Scalars and Vectors Questions with Answers
1) Explain the significance of mathematical quantities in physics.
Physics is the mathematical representation of nature. Almost every principle and concept has a mathematical background. There are countless instances where mathematical techniques are essential to dissect and explain the concepts. The mathematical aspect of the physical concept is the basis of physics. With the help of mathematics, any complex concept can be turned into understandable sets of laws, formulas or equations. The combination of necessary words can describe the motion of bodies. But words cannot explain the complex physics behind motion, and only mathematical methods can decipher and explain these complex concepts consistently. Various mathematical quantities are used to describe physical phenomena. Speed, velocity, acceleration and displacement are described using mathematical quantities.
2) A _____ is a quantity that has only magnitude.
Answer: scalar
Explanation: A scalar is a quantity with only magnitude.
3) A _____ has magnitude as well as a distinct direction.
Answer: vector
Explanation: A vector has magnitude and a distinct direction.
4) What are the main two types of mathematical quantity used in physics?
Mathematical quantities used to represent physical phenomena are divided into two distinct categories: scalar quantity and vector quantity. A scalar is a quantity that has only magnitude, whereas a vector has magnitude and a distinct direction.
5) Explain the fundamentals of a scalar quantity.
In physical science, scalars are physical quantities that only have magnitude. The direction or orientation doesn’t matter in their functionalities, and the magnitude completely denotes them. Some familiar scalar examples are density, energy, mass, time, volume and speed. Complex quantities like force, velocity and acceleration have direction and magnitude (vectors). Scalars are denoted by real numbers that are generally positive and rarely negative. The work done on a body by force is a negative quantity when the body travels in the opposite direction to that the force exerts. Scalars can be calculated using the basic laws of algebra.
6) Explain the fundamentals of a vector quantity.
In mathematics and physics, a vector is a geometric quantity with magnitude and direction, and it is also called a spatial vector or Euclidean vector or geometric vector. A vector can be added using vector algebra. A Euclidean vector is often denoted by a directed line shape or pictorially as an arrow joining a beginning point A with an ending point B.
7) Explain the significance of vectors in physics.
Vectors play a crucial role in physics. The acceleration and velocity of a body and its forces can all be represented by vectors. Numerous other physical quantities can be considered as vectors. Even though most of them do not denote distance, their directions and magnitude can still be denoted by the direction of an arrow and length. The mathematical description of physical vectors depends on the coordinate system employed to describe them.
8) What are the main examples of vectors?
Linear momentum, displacement, acceleration, momentum, angular velocity, force, angular velocity, polarisation, and electric field are the main examples of vectors.
9) What are the main examples of scalars?
Mass, time, energy, volume, speed, and density are the main examples of scalars.
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Practice Questions
1) What is the difference between a scalar and a vector?
2) What is the vector notation for force?
3) Is it possible for a quantity to be neither a scalar quantity nor a vector quantity?
4) How do you denote a vector quantity?
5) What is the vector notation for acceleration?
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