Scalars and Vectors — Class 11 Physics (NCERT 3.1–3.2, 3.3)
1. What Are Scalars and Vectors?
Physics quantities split into two families by one test: how do two of them combine? A scalar is completely specified by a magnitude with its unit and obeys the ordinary arithmetic you already know — mass 2 kg plus mass 3 kg is exactly 5 kg, wherever the masses sit. Distance, speed, time, mass, temperature, work, energy, power, pressure, charge, current and density are all scalars. A vector, by contrast, is specified only when you state its magnitude and its direction, and two vectors combine by the vector addition rules (triangle or parallelogram law), not by arithmetic — a 3 N force east plus a 4 N force north gives 5 N, not 7 N. Displacement, velocity, acceleration, force, momentum, torque and angular momentum are vectors.
This combining test matters because NTA frames classification MCQs around it. Quantities like electric current and pressure carry an associated direction, yet they are scalars, because they add like plain numbers: currents of 3 A and 4 A merging at a junction give 7 A from any geometry. Conversely, the magnitude of any vector — "the speed", "the magnitude of force" — is itself a scalar. Every vector is written as a bold letter or an arrow-topped letter (A or A⃗), with the magnitude |A| called the modulus, always a positive scalar or zero.
2. Complete Theory: Position, Displacement, Null and Unit Vectors
Fix an origin O and axes in the plane. The position vector of a particle P is the vector drawn from O to P. In components it is r = xî + yĵ, where î and ĵ are unit vectors along x and y; its magnitude is r = √(x² + y²). If the particle moves from P₁ (position r₁) to P₂ (position r₂), the displacement is the vector difference Δr = r₂ − r₁ — the straight arrow from P₁ to P₂, independent of the path taken. This is exactly the Chapter 2 idea of displacement, now carrying a second axis; the distance-vs-displacement logic (distance ≥ |displacement|) survives unchanged.
Three special vectors do heavy lifting all chapter. The null (zero) vector 0 has zero magnitude and — importantly — no direction; it arises as A + (−A), and adding it to any vector changes nothing. A unit vector has magnitude 1 and supplies pure direction; any vector can be written as magnitude × direction: A = |A|Â, where  = A/|A| is dimensionless. The basis vectors î, ĵ, k̂ point along +x, +y, +z. Finally, multiplication by a real number λ scales a vector: B = λA has magnitude |λ|·|A|, the same direction as A if λ > 0 and the opposite direction if λ < 0. Two vectors are equal only when both magnitude and direction match, wherever they happen to be drawn — equality is about the arrow, not its location.
3. Visualising the Vector Family
Figure 3.1 — Equal, parallel, antiparallel and collinear vectors
Each small panel shows one relation; the arrow colour only separates the two members of each pair.
Equal: same length, same direction
Parallel: same direction, |lengths| differ
Antiparallel: opposite directions (angle 180°)
Collinear: along one line (any senses)
Exam read-out: "equal vectors" is a two-condition test (magnitude and direction); parallel means θ = 0°, antiparallel means θ = 180°, collinear means both cases at once. In assertion–reason items, "A and −A are equal" is false — equal magnitude but opposite direction.
4. Solved Examples
Example 1 — Displacement between two position vectors
A particle moves from P₁ with position vector r₁ = 2î + 3ĵ (m) to P₂ with r₂ = 5î + 7ĵ (m). Find its displacement vector, its magnitude, and the angle the displacement makes with the +x axis.
Solution (step by step): Displacement is the difference of position vectors: Δr = r₂ − r₁ = (5−2)î + (7−3)ĵ = 3î + 4ĵ (m). Magnitude: |Δr| = √(3² + 4²) = √25 = 5 m. Direction: tan θ = 4/3, so θ = tan⁻¹(4/3) ≈ 53.1° with +x — the signature 3-4-5 triangle at 53°. Note the displacement depends only on endpoints, never on the route the particle actually took.
Example 2 — Classify the quantity
Classify: pressure, work, torque, angular momentum, electric current, displacement.
Solution with reasoning: Scalars: work (force-times-displacement via the dot product, which returns a scalar), pressure (direction of action exists but it adds algebraically), electric current (adds like numbers at a junction — the direction label does not make it a vector). Vectors: torque and angular momentum (defined via cross products, direction by right-hand rule), displacement (straight arrow between endpoints, obeys triangle law). Strategy: ask "how would two such quantities combine?" — the answer decides the family, which is the single most repeated classification logic in NEET.
5. Practice Questions
Q1. For A = 3î + 4ĵ, find |A| and the unit vector along A.
ANSWER: |A| = √(9+16) = 5; Â = A/5 = 0.6î + 0.8ĵ (dimensionless, magnitude 1).
Q2. Find λ so that the vector λî + 2ĵ has magnitude 4.
ANSWER: λ² + 4 = 16 → λ² = 12 → λ = ±2√3 ≈ ±3.46. Both signs are valid — magnitude fixes λ² only.
Q3 (MCQ). Which of the following is a vector? (a) pressure (b) work (c) torque (d) speed
ANSWER: (c) torque — defined by a cross product, direction by the right-hand rule. Pressure, work and speed are scalars.
6. Key Formulas & Takeaways
| Relation | Condition / remark |
|---|---|
| r = xî + yĵ | Position vector; r = √(x² + y²) |
| Δr = r₂ − r₁ | Displacement; endpoint-to-endpoint, path-free |
| Â = A/|A| | Unit vector; dimensionless; needs A ≠ 0 |
| A = |A|Â | Magnitude × direction decomposition |
| B = λA | |λ| scales magnitude; λ < 0 flips direction |
| A + (−A) = 0 | Null vector: zero magnitude, no direction |
Dimension check: components carry the quantity's dimension, unit vectors are dimensionless — so [Ax] = [A] = L for position, consistent with the dimensional method of Chapter 1.
7. Frequently Asked Questions
What is the difference between a scalar and a vector quantity?
A scalar is fully specified by a number (magnitude) with its unit and follows ordinary algebra — mass, speed, work. A vector needs a magnitude, a unit and a direction, and must be added by vector algebra (triangle/parallelogram law), not by simple arithmetic — displacement, force, torque. The real test is how two quantities combine, not whether the quantity 'has a direction'.
Electric current has a direction, so why is it a scalar?
Current has an associated direction but it adds like ordinary numbers: two currents of 3 A and 4 A joining at a junction give 7 A regardless of the geometry, never a 5 A resultant. Because it obeys scalar addition rules, current is classified as a scalar. The same logic applies to pressure, which has a direction of action but combines algebraically.
What is a unit vector and how do you find it?
A unit vector has magnitude exactly 1 and carries only directional information. It is obtained by dividing the vector by its own magnitude:  = A/|A|. It is dimensionless. For example, for A = 3î + 4ĵ, |A| = 5, so  = 0.6î + 0.8ĵ. The standard basis î, ĵ, k̂ are unit vectors along the x, y and z axes.
Can the magnitude of a vector ever be negative?
No. Magnitude is the length of the arrow and is always positive or zero. A negative sign in front of a vector, as in −A, means a vector of the same magnitude pointing in the opposite direction; it does not make the magnitude negative. Multiplying by a scalar λ scales the magnitude by |λ| and reverses direction when λ is negative.
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