Which one of the following is a vector

1. Fundamental and Derived Physical Quantities (basic)

To understand the physical world, we must first learn how to measure it. In physics, any property that can be quantified and measured is called a physical quantity. These are broadly classified into two categories: Fundamental and Derived quantities. Think of fundamental quantities as the 'primary colors' of science; they are independent and cannot be defined in terms of any other quantity. The International System of Units (SI) recognizes seven such base quantities, including Mass, Length, and Time. On the other hand, Derived Quantities are those that are created by mathematically combining fundamental quantities. For instance, Density is a classic derived quantity defined as mass per unit volume. While mass is a fundamental 'brick,' density is a 'structure' built from mass and length (since volume is length cubed) Science, Class VIII (NCERT), Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p.140. This concept is so versatile that we even apply it in geography to understand the density of population, which measures the number of people (a count) relative to the land area INDIA PEOPLE AND ECONOMY, Class XII (NCERT), Population: Distribution, Density, Growth and Composition, p.13. Everything from the speed of a car to the resistivity of a wire Science, Class X (NCERT), Electricity, p.178 is a derived quantity. By mastering these building blocks, we can derive complex relationships, such as Momentum (mass × velocity) or Pressure (force ÷ area). Even in fields like economics, we use derived concepts to calculate a firm's profit by finding the difference between total revenue and total cost Microeconomics (NCERT class XII), The Theory of the Firm under Perfect Competition, p.56. Understanding this hierarchy is the first step toward solving any problem in mechanics.

Feature	Fundamental Quantities	Derived Quantities
Definition	Independent quantities that don't rely on others.	Quantities formulated using fundamental ones.
Examples	Mass (kg), Length (m), Time (s)	Speed (m/s), Density (kg/m³), Force (N)
Number	Only 7 in the SI system.	Virtually unlimited.

Sources: Science, Class VIII (NCERT), Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p.140; Science, Class X (NCERT), Electricity, p.178; INDIA PEOPLE AND ECONOMY, Class XII (NCERT), Population: Distribution, Density, Growth and Composition, p.13; Microeconomics (NCERT class XII), The Theory of the Firm under Perfect Competition, p.56

2. Describing Motion: Distance vs. Displacement (basic)

To understand motion, we must first distinguish between how far an object has traveled and where it ended up relative to where it started. These two concepts are Distance and Displacement. While we often use them interchangeably in daily life, in physics, they describe very different properties of motion.

Distance is the total length of the path traveled by an object. It doesn't matter which way the object turned or if it doubled back; every step counts toward the total. Because distance only cares about "how much ground was covered" and not the direction, it is a scalar quantity. For example, when studying the geography of India, we measure the actual distance from the north to the south extremity as 3,214 km INDIA PHYSICAL ENVIRONMENT, Geography Class XI, India — Location, p.2. Similarly, in experiments involving friction, we observe that an object stops after traveling different distances depending on whether the surface is smooth glass or rough sand Science, Class VIII, Exploring Forces, p.68.

Displacement, on the other hand, is the change in position of an object. It is defined as the shortest straight-line distance between the initial and final points, along with the direction of that line. This makes displacement a vector quantity. A key point for UPSC aspirants to remember is that displacement only depends on the starting and ending coordinates, not the path taken. If you run a full lap around a 400-meter circular track, your distance is 400 meters, but your displacement is zero because you are back where you started.

Feature	Distance	Displacement
Definition	Total path length covered.	Shortest path between start and end.
Type	Scalar (Magnitude only).	Vector (Magnitude + Direction).
Value	Always positive or zero.	Can be positive, negative, or zero.

Sources: INDIA PHYSICAL ENVIRONMENT, Geography Class XI, India — Location, p.2; Science, Class VIII, Exploring Forces, p.68

3. Newton's Laws of Motion (intermediate)

To understand how the universe moves, we must look at Newton’s Laws of Motion, which describe the relationship between an object and the forces acting upon it. At its simplest, a force is a push or a pull resulting from an interaction Science, Class VIII, NCERT (Revised ed 2025), Exploring Forces, p.77. These forces can be contact forces (like friction or muscular force) or non-contact forces (like gravity or magnetism) Science, Class VIII, NCERT (Revised ed 2025), Exploring Forces, p.72.

Newton’s First Law (Law of Inertia) states that an object will remain at rest or in uniform linear motion (constant speed in a straight line) unless an external force acts on it Science-Class VII, NCERT (Revised ed 2025), Measurement of Time and Motion, p.118. This is why a ball rolling on the ground eventually stops; even if we don't "see" a hand stopping it, the force of friction is acting as an external influence to change its state of motion Science, Class VIII, NCERT (Revised ed 2025), Exploring Forces, p.67.

The Second Law provides the mathematical backbone of mechanics: the net force acting on an object is equal to the rate of change of its momentum (F = dp/dt). To master this, we must distinguish between vector quantities (which have both magnitude and direction) and scalar quantities (magnitude only). While mass is a scalar, momentum is a vector because it is the product of mass and velocity. This distinguishes it from Kinetic Energy, which is a scalar representing the 'quantity' of motion without a specific direction. Similarly, density (mass per unit volume) and pressure (force per unit area in fluids) are treated as scalars in general contexts Science, Class VIII, NCERT (Revised ed 2025), How do scientists define density?, p.140.

Quantity	Type	Reasoning
Force	Vector	Has magnitude (Newtons) and a specific direction of push/pull.
Momentum	Vector	The product of mass and velocity (a vector).
Pressure	Scalar	In fluids, it acts equally in all directions at a point.
Density	Scalar	Defined as mass per unit volume; no direction involved.

Finally, the Third Law reminds us that forces always exist in pairs: for every action, there is an equal and opposite reaction. Whether it is the Earth pulling an apple down via gravitational force or the apple pulling the Earth up, the forces are identical in strength but opposite in direction Science, Class VIII, NCERT (Revised ed 2025), Exploring Forces, p.72.

Sources: Science, Class VIII . NCERT(Revised ed 2025), Exploring Forces, p.77; Science, Class VIII . NCERT(Revised ed 2025), Exploring Forces, p.72; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.118; Science, Class VIII . NCERT(Revised ed 2025), Exploring Forces, p.67; Science, Class VIII . NCERT(Revised ed 2025), How do scientists define density?, p.140

4. Work, Energy, and Power (intermediate)

In physics, Work is done only when a force applied to an object causes it to move through a distance. It is mathematically defined as the product of the component of the force in the direction of displacement and the magnitude of this displacement (W = F·d·cosθ). While force and displacement are both vectors, work itself is a scalar quantity because it represents a magnitude of energy transfer without a specific direction. If you push against a stationary wall, despite your effort, the work done is zero because there is no displacement.

Energy is the capacity to do work. It exists in various forms, most notably Kinetic Energy (the energy of motion) and Potential Energy (stored energy). Kinetic energy is a scalar quantity because it represents the total 'quantity of motion' an object possesses, independent of its direction Science, Class VIII, Chapter 9, p.140. In the natural world, kinetic energy is the primary driver of denudational processes; for instance, the erosion and transportation of earth materials by wind, running water, and glaciers are entirely controlled by the kinetic energy these agents possess FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI, Geomorphic Processes, p.43.

Power is the rate at which work is performed or energy is consumed (P = W/t). In the context of national development, power and energy are critical infrastructure components. For example, India is currently the world's third-largest consumer of energy, with a strategy (NEP 2040) focused on making the nation 'Energy Ready' through affordable access and enhanced efficiency Indian Economy, Nitin Singhania, Infrastructure, p.443. While energy tells us the total amount of 'fuel' available, power tells us how quickly we can use that fuel to perform tasks.

Quantity	Type	Definition/Relation
Work	Scalar	Force × Displacement
Kinetic Energy	Scalar	½ × mass × velocity²
Momentum	Vector	Mass × Velocity
Power	Scalar	Work / Time

Sources: Science, Class VIII, Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p.140; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI, Geomorphic Processes, p.43; Indian Economy, Nitin Singhania, Infrastructure, p.443

5. Properties of Matter: Density and Pressure (intermediate)

To understand how matter behaves, we look at two fundamental properties: Density and Pressure. Density is a measure of how 'tightly packed' matter is within a specific space. Defined mathematically as mass per unit volume, it is an intrinsic property of a substance. It is a scalar quantity—it tells us about the concentration of matter without any directional component Science, Class VIII. NCERT (Revised ed 2025), Chapter 9, p. 140.

Pressure, on the other hand, describes how a force is distributed over a surface. It is defined as force per unit area. While the force acting on a surface is a vector, pressure itself is considered a scalar quantity in fluid mechanics because at any given point, it acts equally in all directions Science, Class VIII. NCERT (Revised ed 2025), Chapter 8, p. 82. The Standard International (SI) unit for pressure is the pascal (Pa), which is equivalent to one newton per square metre (N/m²). In practical geography and meteorology, pressure is also expressed in millibars (mb) or hectopascals (hPa), where 1 mb = 1 hPa = 100 Pa Science, Class VIII. NCERT (Revised ed 2025), Chapter 8, p. 87.

In the context of fluids (liquids and gases), pressure is exerted on the walls of the container and increases with depth. Our atmosphere also exerts pressure, known as atmospheric pressure. Variations in this pressure across different regions create a pressure gradient—the rate at which pressure changes over a distance. This gradient is the fundamental force that causes winds; air naturally moves from high-pressure zones to low-pressure zones. On weather maps, closely spaced isobars (lines of equal pressure) indicate a strong pressure gradient and higher wind speeds Fundamentals of Physical Geography, Geography Class XI (NCERT 2025 ed.), Chapter 9, p. 78.

Property	Definition	SI Unit	Type
Density	Mass / Volume	kg/m³	Scalar
Pressure	Force / Area	Pascal (Pa)	Scalar

Sources: Science, Class VIII. NCERT (Revised ed 2025), Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p.140; Science, Class VIII. NCERT (Revised ed 2025), Chapter 8: Pressure, Winds, Storms, and Cyclones, p.82, 87, 94; Fundamentals of Physical Geography, Geography Class XI (NCERT 2025 ed.), Chapter 9: Atmospheric Circulation and Weather Systems, p.78, 84

6. Scalar vs. Vector Quantities (intermediate)

In our study of mechanics, we categorize physical quantities based on the information required to describe them fully. A scalar quantity is defined solely by its magnitude (a numerical value and a unit). For example, density—defined as mass per unit volume—is a scalar because it tells us how 'compact' a substance is without any reference to direction NCERT (Revised ed 2025), Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p.140. Similarly, kinetic energy represents the 'quantity of motion' an object possesses; it doesn't matter if a car is moving North or South, its kinetic energy remains the same if its speed and mass are constant. A tricky example is pressure: while it involves force (a vector), pressure in fluid mechanics is considered a scalar because, at any given point, it acts equally in all directions. Conversely, a vector quantity is only complete when both magnitude and direction are specified. Momentum (p = mv) is a classic vector because it is the product of mass (scalar) and velocity (vector). If you change the direction of an object, you change its momentum, even if its speed stays the same. This distinction is vital when applying Newton’s Second Law, which states that the net force acting on an object is equal to the rate of change of its momentum (F = dp/dt). Since force is a vector, the rate of change of momentum must also be a vector quantity to maintain mathematical consistency.

To help you distinguish them quickly, refer to this comparison:

Feature	Scalar Quantities	Vector Quantities
Definition	Magnitude only	Magnitude + Direction
Examples	Mass, Density, Pressure, Kinetic Energy	Force, Velocity, Momentum, Acceleration
Changes	Changes only when magnitude changes	Changes if magnitude OR direction changes

Sources: NCERT (Revised ed 2025), Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p.140

7. Linear Momentum and Force (Newton's 2nd Law) (exam-level)

To understand motion in the physical world, we must look beyond just 'speed.' We need to understand Linear Momentum (p), which is essentially the 'quantity of motion' an object possesses. It is defined as the product of an object's mass (m) and its velocity (v). Mathematically, we express this as p = mv. Because velocity is a vector quantity (it has both magnitude and direction), momentum is also a vector. This means if you change either the speed or the direction of an object, you are changing its momentum. For instance, when an object is thrown vertically upwards, its speed decreases until it stops momentarily at the top; at that exact point, its momentum is zero before it reverses direction and accelerates downwards Science ,Class VIII . NCERT(Revised ed 2025), Exploring Forces, p.72.

Newton's Second Law of Motion provides the bridge between force and momentum. It states that the net force (F) acting on an object is equal to the rate of change of its momentum with respect to time. This is written as F = dp/dt. If the mass remains constant, this simplifies to the famous F = ma (Force = mass × acceleration). It is crucial for a UPSC aspirant to distinguish between vector quantities like momentum and force, and scalar quantities like pressure or density. While force is a vector, pressure (force per unit area) is considered a scalar in fluid mechanics because it acts equally in all directions at any given point Science ,Class VIII . NCERT(Revised ed 2025), Pressure, Winds, Storms, and Cyclones, p.94.

Quantity	Type	Definition/Context
Linear Momentum	Vector	Product of mass and velocity (p = mv).
Force	Vector	The rate of change of momentum (F = dp/dt).
Pressure	Scalar	Force acting per unit area; acts in all directions Science ,Class VIII . NCERT(Revised ed 2025), Pressure, Winds, Storms, and Cyclones, p.87.
Kinetic Energy	Scalar	Energy of motion; does not have a direction.

Sources: Science ,Class VIII . NCERT(Revised ed 2025), Exploring Forces, p.72; Science ,Class VIII . NCERT(Revised ed 2025), Pressure, Winds, Storms, and Cyclones, p.94; Science ,Class VIII . NCERT(Revised ed 2025), Pressure, Winds, Storms, and Cyclones, p.87

8. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamental definitions of physical quantities, this question tests your ability to move beyond simple memorization and apply your conceptual building blocks. The core challenge here is distinguishing between scalar quantities, which only possess magnitude, and vector quantities, which require both magnitude and direction to be fully defined. As you learned in Science, Class VIII, NCERT (Revised ed 2025), understanding how basic properties like mass and velocity interact is the key to identifying these derived physical quantities accurately.

To arrive at the correct answer, let’s walk through the reasoning using Newton’s Second Law of Motion. You know that momentum is the product of mass (a scalar) and velocity (a vector), which makes momentum itself a vector. The rate of change of momentum is the very definition of net force acting on an object. Since force necessarily acts in a specific direction to cause acceleration, the rate of change of momentum must be a vector. This logical chain—linking velocity to momentum and momentum to force—is the exact mental framework you need to solve UPSC's conceptual physics problems.

It is equally important to recognize why the other options are classic UPSC traps. Pressure is the most common pitfall; students often assume it is a vector because it involves force, but in fluid mechanics, it is a scalar because it acts equally in all directions at a point, according to NASA GRC (Vectors). Similarly, Kinetic energy represents a magnitude of motion without directional dependence, and Density is merely a ratio of two scalars (mass and volume). Remember: just because a quantity is derived from a vector doesn't mean the final result is one!