Q.Define vector product. Explain the properties of a vector product with two examples.
🔒You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.
🔒 Start your 14-day free trial to unlock the full solution →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.
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.
A vector is often represented as A or in bold as A. Its magnitude is written as ∣A∣ or simply 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.
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. …
The vector (cross) product of two vectors gives a new vector perpendicular to both, and it has several characteristic properties. …
Vector product A x B = AB sin(theta) n is a vector perpendicular to both. Properties include: anti-commutative (A x B = -B x A, e.g. i x j = k, j x i = -k) and zero for parallel vectors (i x i = 0); it is also distributive over addition.
Definition of vector product:
The vector product (cross product) of two vectors A and B is defined as:
A x B = A B sin(theta) n
where A and B are the magnitudes, theta is the angle between them (0 to 180 degrees), and n is a unit vector perpendicular to the plane containing A and B, with direction given by the right-hand rule. The result is itself a vector.
Properties of the vector product (with examples):
-
It is not commutative; it is anti-commutative:
A x B = -(B x A)
Reversing the order reverses the direction of the resultant.
Example: For unit vectors, i x j = k, but j x i = -k.
-
The vector product of two parallel (or anti-parallel) vectors is zero:
If theta = 0 or 180 degrees, sin(theta) = 0, so A x B = 0. In particular a vector's product with itself is zero. …
Showing the 12 most recent of 23 on this concept.
- CBSE 2026Set ANNUAL1 markQ.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. …
- CBSE 2026Set ANNUAL1 markMCQQ.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. …
- CBSE 2026Set ANNUAL1 markMCQQ.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 …
- CBSE 2025Set ANNUAL1 markMCQQ.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. …
- CBSE 2024Set ANNUAL1 markMCQQ.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. …
- CBSE 2024Set ANNUAL1 markQ.A physical quantity having both magnitude and direction is called ______.
›Reveal solutionSolution
[!TLDR]
Vector
Method
By definition of a vector qua …
- CBSE 2023Set 55/3/11 markMCQQ.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. …
-
- CBSE 2023Set ANNUAL1 markMCQQ.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. …
- CBSE 2023Set ANNUAL1 markMCQQ.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.
…
- CBSE 2023Set ANNUAL1 markMCQQ.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).
…
- CBSE 2022Set TERM11 markMCQQ.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). …
- CBSE 2022Set ANNUAL1 markMCQQ.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). …
🎓Unlock everything free for 14 days
- ✓Full step-by-step solutions
- ✓Concept-first explanations
- ✓Methods, shortcuts & mistakes
- ✓PYQ mapping + timed mock tests
Full access for 14 days. No credit card required.