Read each statement below carefully and state, with reasons and examples, if it is true or false: 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 space, (e) has the same value for observers with different orientations of axes.

> [!TLDR] > A scalar is defined by its invariance under coordinate rotations — only statement (e) is true. The other four statements describe properties that are neither necessary nor sufficient for a quantity to be a scalar. The question asks you to test each statement against the actual definition of a scalar. This is a classic trap: most students confuse "scalar" with "magnitude" or "constant" or "dimensionless number." Let's clear that up first. A scalar is a quantity that is completely specified by a single number (with appropriate units) and — this is the crucial part — remains unchanged when you rotate your coordinate axes. Temperature is a scalar; the x-component of velocity is not, because if you rotate your axes, that component changes. The number itself can be positive, negative, zero, have dimensions, vary in space, and may or may not be conserved. None of those matter for the definition. Now examine each statement one by one. 1. **Statement (a): "A scalar quantity is one that is conserved in a process."** This is false. Conservation is a property of certain quantities in certain physical processes (like energy or charge), but it has nothing to do with being a scalar. For example, mass is a scalar, but it is not conserved in nuclear reactions — it can convert to energy. Conversely, momentum is a vector, yet it is conserved in isolated systems. The definition of a scalar does not involve conservation at all. 2. **Statement (b): "A scalar quantity can never take negative values."** False. Scalars can absolutely be negative. Temperature in Celsius can be -10°C. Electric potential can be -5 V. Work done by friction can be negative. The sign of a scalar simply indicates a direction along a one-dimensional scale (like "below zero" or "opposite to the chosen reference"), not a vector direction. The common confusion arises because "magnitude" is always non-negative, but a scalar is not the same as its magnitude. 3. **Statement (c): "A scalar quan

Read each statement below carefully and state, with reasons and examples, if it is true or false: A scalar qua

Exercises · 3.21

Q. Read each statement below carefully and state, with reasons and examples, if it is true or false: A scalar quantity is one that

(a) is conserved in a process,

(b) can never take negative values,

(d) does not vary from one point to another in space,

(e) has the same value for observers with different orientations of axes.

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Concept understanding — Scalar vs Vector

Scalar vs Vector: From Intuition to Precision

Imagine you're telling a friend about your day. You might say, "It was 35 degrees outside." That's a complete piece of information — just a number with a unit. Now imagine saying, "I walked 5 kilometres." That's also a number with a unit, but something feels incomplete. Where did you walk? Did you go north? East? In circles? The "5 kilometres" alone doesn't tell the full story.

This difference is the entire idea behind scalars and vectors.

The Intuition

Scalars are quantities that are fully described by a magnitude (a number and a unit). Temperature, mass, time, speed, energy — these are scalars. If I say "the mass is 10 kg," you know everything there is to know about that mass. There's no direction to mass.

Vectors are quantities that need both magnitude and direction to be fully described. Displacement, velocity, force, acceleration — these are vectors. "5 km north" is a vector. "10 N downward" is a vector. The direction is not optional; it's part of the quantity itself.

Note

Speed is a scalar (just "how fast"). Velocity is a vector ("how fast" + "in which direction"). A car moving at 60 km/h has a speed of 60 km/h. If it's moving east at 60 km/h, its velocity is 60 km/h east.

The Precise Statement

A scalar is a physical quantity that has only magnitude. It obeys ordinary arithmetic: you can add, subtract, multiply, and divide scalars just like numbers.

A vector is a physical quantity that has both magnitude and direction. Vectors obey special rules of addition (like the triangle law or parallelogram law) because direction matters.

\vec{A}

Key Differences at a Glance

Property	Scalar	Vector
Description	Magnitude only	Magnitude + Direction
Example	25°C, 10 kg, 5 seconds	10 m/s north, 20 N downward
Addition	Ordinary arithmetic	Triangle/parallelogram law
Division by a scalar	Yes (e.g., distance ÷ time = speed)	Yes (e.g., displacement ÷ time = velocity)
Division by a vector	Not defined	Not defined

Why This Matters

When you add two scalars, say 5 kg + 3 kg, you get 8 kg. Simple.

When you add two vectors, say walking 3 km east then 4 km north, you don't get 7 km. You get 5 km northeast (by Pythagoras). The direction of each step matters. If you walked 3 km east and then 4 km west, you'd end up 1 km east — not 7 km anywhere.

Watch out

A common mistake is treating vectors like scalars in addition. Never simply add magnitudes of vectors unless they point in exactly the same direction. Always account for direction.

The Mathematical Representation

In one dimension, we often use a sign to indicate direction: $+5$ m/s means right, $-5$ m/s means left. The sign is the direction. …

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A scalar is defined by its invariance under coordinate rotations — only statement (e) is true. The other four statements describe properties that are neither necessary nor sufficient for a quantity to be a scalar.

The question asks you to test each statement against the actual definition of a scalar. This is a classic trap: most students confuse "scalar" with "magnitude" or "constant" or "dimensionless number." Let's clear that up first.

A scalar is a quantity that is completely specified by a single number (with appropriate units) and — this is the crucial part — remains unchanged when you rotate your coordinate axes. Temperature is a scalar; the x-component of velocity is not, because if you rotate your axes, that component changes. The number itself can be positive, negative, zero, have dimensions, vary in space, and may or may not be conserved. None of those matter for the definition.

Now examine each statement one by one.

Statement (a): "A scalar quantity is one that is conserved in a process."

This is false. Conservation is a property of certain quantities in certain physical processes (like energy or charge), but it has nothing to do with being a scalar. For example, mass is a scalar, but it is not conserved in nuclear reactions — it can convert to energy. Conversely, momentum is a vector, yet it is conserved in isolated systems. The definition of a scalar does not involve conservation at all.
Statement (b): "A scalar quantity can never take negative values."

False. Scalars can absolutely be negative. Temperature in Celsius can be -10°C. Electric potential can be -5 V. Work done by friction can be negative. The sign of a scalar simply indicates a direction along a one-dimensional scale (like "below zero" or "opposite to the chosen reference"), not a vector direction. The common confusion arises because "magnitude" is always non-negative, but a scalar is not the same as its magnitude.
Statement (c): "A scalar quantity must be dimensionless."

False. Most scalars in physics have dimensions. Temperature has dimensions of kelvin, mass has dimensions of kilograms, time has seconds, energy has joules. Dimensionless quantities (like refractive index or strain) are a special subset of scalars, but they are not the whole set. A scalar can carry any physical dimension.
Statement (d): "A scalar quantity does not vary from one point to another in space."

False. Scalars can be functions of position. Temperature varies from place to place — that's why we have weather maps. Pressure varies with altitude. Electric potential varies around a charge. A scalar field is precisely a quantity that assigns a scalar value to every point in space. Variation in space is perfectly allowed.
Statement (e): "A scalar quantity has the same value for observers with different orientations of axes." …

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Concept: A Scalar Is Whatever Survives a Rotation of Axes Unchanged

Method: State the Rotation-Transformation Law Explicitly, Then a Compact Counterexample Table (Fresh Examples, Not Temperature/Mass)

The existing solutions work through each statement in flowing prose using temperature, mass, and nuclear-reaction examples. This method instead starts from the formal transformation rule that distinguishes a scalar from a vector component, and then disposes of the four false statements with one compact table using a different set of concrete counterexamples.

The formal test

Under a rotation of coordinate axes by angle $\theta$ , a vector's components mix:

A_x' = A_x\cos\theta+A_y\sin\theta, \qquad A_y'=-A_x\sin\theta+A_y\cos\theta

A scalar $S$ , by contrast, has only one number to begin with — there is nothing for it to mix with, and the transformation is trivially $S'=S$ for every $\theta$ . This is not an extra property scalars happen to have; it is the operational definition of what "scalar" means. So statement (e) is true by this transformation law directly.

Compact counterexample table for the other four (fresh examples)

Statement	Counterexample	Verdict
(a) conserved in a process	Kinetic energy is a scalar, yet is not conserved in an inelastic collision (some converts to heat/sound)	False
(b) never negative	Gravitational potential energy $U=-\dfrac{GMm}{r}$ is a scalar that is always negative (with the usual zero-at-infinity convention)	False
(c) must be dimensionless	Entropy is a scalar with real dimensions, $\text{J/K}$ — not a pure number	False
(d) same at every point in space	Electrostatic potential $V$ around a point charge is a scalar field — it genuinely varies with distance from the charge	False

Why the counterexamples work …

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