State, for each of the following physical quantities, if it is a scalar or a vector: volume, mass, speed, acce

Exercises · 3.1

Q. State, for each of the following physical quantities, if it is a scalar or a vector: volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity.

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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.

In two or three dimensions, we use components. A vector $\vec{v}$ can be written as:

\vec{v} = v_x \hat{i} + v_y \hat{j} + v_z \hat{k}

where $v_x, v_y, v_z$ are the components (scalars) along the $x, y, z$ axes, and $\hat{i}, \hat{j}, \hat{k}$ are unit vectors (vectors of magnitude 1) pointing along those axes.

Important

A unit vector has magnitude 1 and gives direction. Any vector can be written as its magnitude times a unit vector in its direction: $\vec{A} = |\vec{A}| \,\hat{a}$ .

The Bottom Line

Scalar: "How much?" — just a number with a unit.

Vector: "How much and which way?" — a number with a unit and a direction.

Every time you encounter a physical quantity, ask yourself: does direction matter here? If yes, it's a vector. If no, it's a scalar. This simple question will guide you through physics, from kinematics to electromagnetism.

"Scalar vs Vector derivation" and "Scalar vs Vector numerical problems" are two of the most common searches tied to this topic, and Scalar vs Vector is drawn directly from the Motion in a Plane coverage of the NCERT/CBSE Class 11 Physics syllabus and recurs often in JEE Main and NEET papers. Pairing this explanation with NCERT Physics textbook practice and previous years' questions is the surest way to lock the concept in before an exam.

The classification hinges on whether the quantity has direction (vector) or only magnitude (scalar). Volume, mass, speed, density, number of moles, and angular frequency are scalars. Acceleration, velocity, displacement, and angular velocity are vectors.

The core idea is simple: a scalar is fully described by a number (and its unit), while a vector needs both a magnitude and a direction in space. Many quantities in physics are obviously one or the other, but a few — like angular frequency — can trick you if you only go by name.

Let’s go through each quantity one by one.

Volume — Scalar.

Volume tells you how much space an object occupies. “5 litres” is a complete description; there is no direction associated with “how much space”. Even if you pour water into a tilted flask, the volume is still just a number.
Mass — Scalar.

Mass is a measure of the amount of matter. “2 kg” is all you need. Direction is irrelevant.
Speed — Scalar.

Speed is the rate of change of distance, not displacement. “The car moves at 60 km/h” gives no information about which way it is going. That’s why speed is the magnitude of velocity, and magnitude alone is scalar.
Acceleration — Vector.

Acceleration is the rate of change of velocity. Since velocity itself is a vector, changing it requires a direction. For example, “9.8 m/s² downward” is a complete description. Even if an object is slowing down, the acceleration vector points opposite to the velocity.
Density — Scalar.

Density is mass per unit volume. “The density of water is 1000 kg/m³” — no direction needed. It’s a property of the material, not of motion.
Number of moles — Scalar.

This is a count of particles (Avogadro’s number of entities). “2 moles of oxygen” is a pure number; direction has no meaning.
Velocity — Vector.

Velocity is speed with direction. “5 m/s east” is a vector. If you only say “5 m/s”, you’ve given speed, not velocity. This is the classic distinction.
Angular frequency — Scalar.

This is a tricky one. Angular frequency $\omega$ tells you how many radians per second an object rotates through. It is a scalar because it describes a rate of angle change, not a direction in space.

Watch out

Do not confuse angular frequency with angular velocity. Angular velocity is a vector (pointing along the axis of rotation by the right-hand rule). Angular frequency is just its magnitude — a scalar. In equations like $x = A \cos(\omega t)$ , $\omega$ is a positive number.
Displacement — Vector.

Displacement is the straight-line distance from the starting point to the ending point, with direction. “3 km north” is a displacement. If you walk in a circle and return home, your displacement is zero — the direction is undefined, but the concept is still vectorial.
Angular velocity — Vector.

Angular velocity $\vec{\omega}$ has magnitude (rad/s) and a direction along the axis of rotation (given by the right-hand rule). For a spinning wheel, the vector points perpendicular to the plane of rotation. This is a true vector.

Tip

A quick memory aid: if the quantity can be negative in a 1D problem and that negative sign means “opposite direction” (like velocity = -5 m/s), it’s a vector. If a negative sign just means “less than zero” (like temperature), it’s a scalar. Here, acceleration, velocity, displacement, and angular velocity all use the sign for direction.

✓Final answer

The scalars are: volume, mass, speed, density, number of moles, angular frequency. The vectors are: acceleration, velocity, displacement, angular velocity.

Concept: A Two-Question Test, Applied Uniformly

Method: Systematic Definitional Test (Table), Not Item-by-Item Narrative

Rather than reasoning through each quantity's physical meaning in prose, this method applies exactly the same two-question test to every entry and records the verdict in a table -- turning ten separate judgment calls into one repeated, mechanical check.

The test

Q1: Does the quantity need a direction in space to be completely specified (not just "which way it flows/rotates" loosely, but genuine spatial direction)?
Q2 (tie-breaker only, for quantities that sound directional but aren't): Does it combine with others of its kind by the parallelogram/vector-addition law, or by plain arithmetic?

If Q1 is "no", it's a scalar. If Q1 is "yes" and Q2 confirms vector addition applies, it's a vector.

Applying the test to each quantity

Quantity	Q1: needs direction?	Verdict
Volume	No -- a single number (litres) fully describes it	Scalar
Mass	No -- amount of matter, no direction	Scalar
Speed	No -- magnitude of velocity only	Scalar
Acceleration	Yes -- rate of change of the vector velocity	Vector
Density	No -- mass/volume, a ratio of two scalars	Scalar
Number of moles	No -- a count	Scalar
Velocity	Yes -- speed with a direction of travel	Vector
Angular frequency $\omega$	No -- describes a rate of angle change (a number of rad/s), not a direction in space; unlike angular velocity it carries no axis	Scalar
Displacement	Yes -- straight-line change of position, with direction	Vector
Angular velocity	Yes -- magnitude (rad/s) and an axis direction, via the right-hand rule	Vector

Why the tabulated test is worth doing

The one genuinely tricky pair here is angular frequency vs. angular velocity -- both "sound" rotational, but only angular velocity carries an actual spatial axis-direction that obeys vector addition (two rotations about different axes add vectorially); angular frequency is just the numeric rate. Running every entry through the same explicit test, rather than reasoning fresh each time, is what catches this distinction reliably instead of by memorized special-casing.

Final Answer

Scalars: volume, mass, speed, density, number of moles, angular frequency.

Vectors: acceleration, velocity, displacement, angular velocity.

Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

Showing the 12 most recent of 22 on this concept.

CBSE 2026Set ANNUAL1 mark
$Q. When are the two vectors said to be equal vectors?$
›Reveal solutionSolution
Equal vectors must match in both magnitude and direction — their points of application don't matter.

Two vectors A and B are said to be equal vectors if and only if:

They have the same magnitude, |A| = |B|.

They point in the same direction.

If both conditions hold, the vectors are equal even though they may be drawn starting from different points in space (a vector, being a free vector, can be translated parallel to itself without changing its identity). In component form, two vectors are equal if their corresponding components along each axis are equal (Ax = Bx, Ay = By, Az = Bz).

✓Final answer
$Two vectors are equal when they have equal magnitude and identical direction, irrespective of their location/starting point.$
CBSE 2026Set ANNUAL1 markMCQ
$Q. The angle between two equal vectors is :$
(a) 0°
(b) 30°
(c) 90°
(d) 180°
›Reveal solutionSolution
Equal vectors point in exactly the same direction, so the angle between them is 0°.

Two vectors A and B are said to be equal (A = B) only when:

They have the same magnitude, |A| = |B|, and

They point in the same direction.

Since both vectors point the same way, the angle between their directions is 0°. This is different from two vectors that merely have equal magnitude but different directions — those are NOT called equal vectors.

✓Final answer
$The correct option is (a) 0°.$
CBSE 2026Set ANNUAL1 markMCQ
$Q. Which of the following is not a vector quantity ?$
(a) Velocity
(b) Displacement
(c) Speed
(d) Impulse
›Reveal solutionSolution
Speed is a scalar; the other three are vectors. Answer (C).

A vector quantity has both magnitude and direction; a scalar has magnitude only.

Velocity -> has direction -> vector

Displacement -> has direction -> vector

Impulse (= change in momentum) -> has direction -> vector

Speed -> the magnitude of velocity, no direction -> scalar

✓Final answer
$(C) Speed.$
CBSE 2025Set ANNUAL1 markMCQ
$Q. Which of the following quantities is a vector? (A) Temperature (B) Electric current (C) Impulse (D) Volume$
›Reveal solutionSolution
Impulse is a vector because it equals the change in the vector quantity momentum.

Check each option:

Temperature: has only magnitude — scalar.

Electric current: though it has a direction associated with conventional flow, it does not obey the vector addition law — it is treated as a scalar in physics.

Impulse: $\vec{J} = \vec{F}\,\Delta t = \Delta \vec{p}$ , the change in momentum. Since momentum is a vector, impulse is a vector, pointing along the direction of the applied force.

Volume: has only magnitude — scalar.

✓Final answer
$(C) Impulse.$
CBSE 2024Set ANNUAL1 markMCQ
$Q. Which of the following quantities is a vector?$
(a) Speed
(b) Distance
(c) Velocity
(d) Time
›Reveal solutionSolution
Velocity is the only vector in the list because it needs both a magnitude and a direction to be fully specified.

Speed: only magnitude (e.g. 10 m/s) — scalar.

Distance: only magnitude (total path length) — scalar.

Velocity: magnitude AND direction (e.g. 10 m/s due east) — vector.

Time: only magnitude — scalar.

✓Final answer
$(c) Velocity.$
CBSE 2023Set 55/3/11 markMCQ
$Q. Which one of the following is not a scalar quantity ?$
(a) Electric field
(b) Voltage
(c) Resistivity
(d) Power
›Reveal solutionSolution
The key idea is that scalar quantities have only magnitude, while vector quantities have both magnitude and direction. Electric field is a vector; voltage, resistivity, and power are scalars. So the answer is (a) Electric field.

The Concept: Scalars vs. Vectors

In physics, quantities are classified as either scalars or vectors. A scalar is fully described by its magnitude (a number and a unit) — think of temperature, mass, or energy. A vector requires both magnitude and direction — think of force, velocity, or displacement. The question asks you to spot the odd one out: which of these four is not a scalar? That means it must be a vector.

Let’s examine each option carefully.

Step-by-Step Reasoning

Electric field (option a)

The electric field at a point is defined as the force per unit positive charge placed there. Force is a vector, so electric field inherits that direction. It points away from positive charges and toward negative ones. You can’t fully describe an electric field without saying which way it points.

→ Electric field is a vector quantity.

Voltage (option b)

Voltage, also called electric potential difference, is the work done per unit charge to move a charge between two points. Work is a scalar (it has no direction in space), so voltage is a scalar. You can say “12 volts” without specifying a direction.

→ Voltage is a scalar quantity.

Resistivity (option c)

Resistivity is a material property that tells you how strongly it opposes current. It depends only on the material and its temperature, not on the shape or orientation of the sample. There’s no direction associated with it.

→ Resistivity is a scalar quantity.

Power (option d)

Power is the rate at which energy is transferred or converted. Energy and time are both scalars, so power is a scalar. In electrical contexts, $P = VI$ gives a number (watts) with no direction.

→ Power is a scalar quantity.

Watch out

A common mistake is to think that because current has a direction, electrical quantities like voltage or power must also be vectors. But current itself is a scalar (it’s the rate of flow of charge, and direction is handled separately by the sign). Voltage is energy per charge — energy has no direction. Always check the definition: if the quantity comes from work or energy, it’s almost certainly a scalar.

Final Answer

✓Final answer
$The quantity that is not a scalar is (a) Electric field .$
CBSE 2023Set ANNUAL1 markMCQ
$Q. Which of the following is NOT a property of the null (zero) vector?$
(a) A→ + 0→ = A→
(b) 0·A→ = 0→
(c) λ·0→ = 0→
(d) A→ + A→ = 0→
›Reveal solutionSolution
Options (a), (b), (c) are standard, always-true properties of the null vector; option (d) is not — it only holds if A→ itself happens to be the null vector.

The null vector 0→ is defined as a vector of zero magnitude with no specific direction. Its standard properties are:

A→ + 0→ = A→ — adding the null vector to any vector leaves it unchanged (0→ is the additive identity for vectors). Always true.

0 · A→ = 0→ — multiplying any vector by the scalar 0 gives the null vector. Always true.

λ · 0→ = 0→ — multiplying the null vector by any scalar λ gives the null vector. Always true.

A→ + A→ = 0→ — this says 2A→ = 0→, which is only possible if A→ itself is the null vector. For a general (nonzero) vector A→, A→ + A→ = 2A→ ≠ 0→. So this is NOT a general property of the null vector.

✓Final answer
$The correct option is (d) A\to + A\to = 0\to — it is not a property of the null vector for a general vector A\to (the true relation A\to - A\to = 0\to holds instead).$
CBSE 2023Set ANNUAL1 markMCQ
$Q. Which one of the following physical quantities cannot be represented by a scalar ?$
(a) Momentum
(b) Mass
(c) Magnitude of acceleration
(d) Length
›Reveal solutionSolution

Momentum is a vector (it has direction, since it equals mass times the vector velocity); the other three listed quantities are scalars.

A scalar quantity is completely described by its magnitude alone (with appropriate units) -- examples: mass, length, time, magnitude of acceleration, speed, temperature, energy.

A vector quantity needs both magnitude and direction -- examples: displacement, velocity, acceleration, force, momentum.

Momentum is defined as p = m v, where v is velocity, a vector. Multiplying a vector by a scalar (mass) still gives a vector, so momentum has a direction (the same direction as the velocity) and cannot be represented by a scalar alone.

Mass, magnitude of acceleration (the magnitude of any vector is always a scalar), and length are all scalar quantities.

✓Final answer
$The correct option is (a) Momentum.$
CBSE 2023Set ANNUAL1 markMCQ
$Q. Which of the following is not vector quantity?$
(a) Displacement
(b) Force
(c) Acceleration
(d) Speed
›Reveal solutionSolution

Speed is a scalar, so it is not a vector quantity.

A vector quantity has both magnitude and direction (displacement, force, acceleration, velocity). A scalar has magnitude only (speed, mass, time, distance).

Among the options, displacement, force and acceleration are vectors; speed is the magnitude of velocity and has no direction.

✓Final answer
$(D) Speed.$
CBSE 2022Set TERM11 markMCQ
$Q. Which of the following is not a scalar quantity?$
(1) Temperature
(2) Coefficient of friction
(3) Charge
(4) Impulse
›Reveal solutionSolution

A scalar quantity has magnitude only; impulse, being equal to the vector change in momentum (a vector force integrated over time), has both magnitude and direction and is therefore a vector.

Temperature -- has only magnitude (a number with a unit), no direction: scalar.

Coefficient of friction -- a pure (dimensionless) ratio between two force magnitudes: scalar.

Charge -- has magnitude and sign (+/-), but no spatial direction; treated as scalar (it adds algebraically, not by vector addition).

Impulse -- J = F.delta t = delta p (change in momentum). Since momentum p = mv is a vector, impulse inherits direction from the force/velocity change, and combines by vector addition. It is a vector.

✓Final answer
$(4) Impulse.$
CBSE 2022Set ANNUAL1 markMCQ
$Q. Which of the following is scalar quantity?$
(a) Displacement
(b) Velocity
(c) Work
(d) Force
›Reveal solutionSolution
Work is a scalar (only magnitude); displacement, velocity and force are all vectors.

Concept. A scalar quantity is completely described by a magnitude (a number with a unit) alone, with no associated direction. A vector quantity needs both magnitude and direction.

Checking each option:

Displacement — has magnitude and direction (vector).

Velocity — rate of change of displacement, has direction (vector).

Work $= \vec F \cdot \vec d$ (a dot product of two vectors) — the dot product of two vectors is always a scalar; work has only magnitude (joules), no direction.

Force — has magnitude and direction (vector).

✓Final answer
$Work is a scalar quantity.$
CBSE 2022Set ANNUAL1 mark
$Q. If a vector C̄ is multiplied by a scalar n, what will be the resultant vector.$
›Reveal solutionSolution
Scalar multiplication scales a vector's magnitude by |n| and flips its direction if n is negative.

Rule. If a vector $\bar C$ is multiplied by a scalar $n$ , the result is a new vector $n\bar C$ such that:

Magnitude: $|n\bar C| = |n|\,|\bar C|$ — the length is scaled by the factor $|n|$ .

Direction: same as $\bar C$ if $n>0$ ; exactly opposite (reversed) if $n<0$ ; the zero vector if $n=0$ .

So, for example, $2\bar C$ is a vector twice as long as $\bar C$ in the same direction, while $-\bar C$ has the same length but points the opposite way.

✓Final answer
$The resultant of multiplying vector C̄ by scalar n is the vector nC̄: magnitude n|C̄| (for n>0), same direction as C̄ if n is positive, opposite direction if n is negative.$

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Q.State, for each of the following physical quantities, if it is a scalar or a vector: volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity.

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