Vectors and Scalars Notes — CBSE Class 11 & NEET Guide

 - Dr.Sanjaykumar Pawar  

Vectors are drawn as arrows — length shows magnitude, arrowhead shows direction.

 

Internal Links

	Links To
parallelogram law of vector addition	Parallelogram Law — Class 11 Physics Notes
resolution of vectors into components	Resolution of Vectors & Rectangular Components
dot product and cross product	Scalar and Vector Product of Two Vectors
motion in a plane	Motion in a Plane — Full Chapter Notes (NCERT Ch. 4)
projectile motion numericals	Projectile Motion — Solved Problems for NEET
CBSE Class 11 Physics MCQs	Class 11 Physics — Chapter-wise MCQ Bank
NEET previous year vector questions	NEET PYQs — Vectors & Motion in a Plane

NEET Physics · Chapter 2

Vectors & Scalars

The quantities that need a direction, the ones that don't, and the one rule that tells them apart. Complete beginner notes with worked diagrams.

THE TUBE-AND-BALL PROBLEM — a right triangle hiding in a physics question

01 — Foundations

Why physics needs vectors

Mathematics is the language of physics. Some quantities are fully described by just a number. Others refuse to make sense without a direction attached. Splitting these two apart is the entire point of this chapter — and it shows up in almost every numerical on the NEET paper afterward, from projectile motion to electric fields.

02 — The simple ones

Scalars

Definition: quantities completely described by a numerical value (with a unit) alone. No direction is involved, and they combine using ordinary algebra.

Worked example

A system made of two bodies — one of mass 5 kg, the other 2 kg — has a combined mass of:

5 kg + 2 kg = 7 kg

No angles, no diagrams. Just addition. That is the signature of a scalar.

Common scalars: mass, time, temperature, speed, energy, work, power, distance, charge, density.

03 — The directional ones

Vectors

Definition: quantities that need both magnitude and direction for a complete description, and which add according to the geometric triangle law — not plain algebra.

representation

FIG. A — vectors drawn to scale: 1 cm ≡ 1 m/s (arbitrary chosen scale)

Anatomy of a vector arrow

Part	Name	Meaning
Back end	Tail	Starting point
Front end	Head	Points in the direction of the vector
Length	Magnitude	Numerical size, drawn to scale

Notation: written with an arrow on top — $\vec{AB}$, $\vec{v}$ — or in bold print: AB, v, F.

TRAP

Having a direction is not enough. Electric current flows through a wire in a direction — but current does not add up by the triangle rule. Two currents meeting at a junction just add algebraically (Kirchhoff's rule), not geometrically. So current is a scalar, despite having a direction. This exact question appears repeatedly in NEET-level papers.

04 — The addition rule

Triangle law of vector addition

Statement: if two vectors are represented, in magnitude and direction, by two sides of a triangle taken in order, the resultant is given by the third side, taken in the reverse order.

step 1

Draw the first vector, A to B

step 2

From B, draw the second vector, B to C

step 3 — the resultant

Join A to C. This is $\vec{AB} + \vec{BC} = \vec{AC}$

RULE

The resultant always runs from the tail of the first vector to the head of the last vector — regardless of how many vectors you chain together, tail-to-head.

05 — Solved numerical

The tube-and-ball problem

A small ball moves inside a long tube at 3 m/s while the tube itself moves at 4 m/s perpendicular to its own length. What is the ball's resultant velocity, as seen from the room?

Pythagoras hiding inside a physics question: 3-4-5 triangle

Solution

In 1 second: tube carries the ball 3 m along its length; the ball also moves 4 m perpendicular to it. The two displacements form a right triangle, so:

R = √(3² + 4²) = √25 = 5 m

Since this happened in 1 s, the resultant velocity is 5 m/s, directed at

θ = tan⁻¹(4/3) = 53°

…from the direction of the tube.

MEMORIZE

The 3–4–5 right triangle (and its cousin 5–12–13) shows up constantly in NEET vector numericals. Spotting it instantly saves calculator time in the exam.

06 — Reference

Quick formula box

Concept	Formula
Resultant magnitude	R = √(A² + B² + 2AB cos θ)
Direction of resultant	tan α = B sin θ / (A + B cos θ)
Maximum resultant (θ = 0°)	R = A + B
Minimum resultant (θ = 180°)	R = |A − B|
Perpendicular vectors (θ = 90°)	R = √(A² + B²)

This is the parallelogram-law version of the same triangle rule — NEET numericals usually hand you this formula directly, so both pictures are worth knowing.

07 — Side by side

Scalar vs vector

Property	Scalar	Vector
Needs	Magnitude + unit only	Magnitude + unit + direction
Addition rule	Ordinary algebra	Triangle / parallelogram law
Examples	mass, speed, work, energy, current, charge	displacement, velocity, acceleration, force, momentum
Sign	Can be negative (e.g. temperature)	Magnitude always positive; direction shows sense

Last-minute recall

• Scalar = magnitude + unit. Vector = magnitude + unit + direction + triangle law.
• Triangle law: tail-to-head arrangement; resultant runs from the very first tail to the very last head.
• Current has a direction but is still a scalar — it fails the triangle law test.
• 3-4-5 right triangle → resultant 5, angle 53°. A recurring numerical pattern — recognize it on sight.
• Vector addition is commutative: $\vec{A}+\vec{B} = \vec{B}+\vec{A}$.

NEET PHYSICS — VECTORS & SCALARS · FIELD NOTES

CBSE · Class 11 Physics · Ch. Vectors & Scalars

Complete Question Bank

Every CBSE exam format in one place — MCQs, assertion-reason, fill-in-the-blanks, match-the-column, case study, and short/long answers. Tap any question to reveal the answer.

Very Short Short Answer Long Answer MCQs Assertion-Reason Fill in Blanks Match Column Statement-Based Case Study

Section A

Very Short Answer Questions

1 mark each — one line / one word answers

1 markQ1. Define a scalar quantity.

Show answer

A scalar is a physical quantity that is completely described by its magnitude (with a proper unit) alone; it has no associated direction. Example: mass, time.

1 markQ2. Define a vector quantity.

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A vector is a physical quantity that requires both magnitude and direction for its complete description, and which obeys the triangle law of addition.

1 markQ3. Give one example each of a scalar and a vector quantity other than mass and velocity.

Show answer

Scalar: electric charge (or work, energy). Vector: force (or momentum, acceleration).

1 markQ4. Is electric current a vector quantity? Justify in one line.

Show answer

No. Although current has a direction of flow, it does not add according to the triangle law of vector addition, so it is treated as a scalar.

1 markQ5. What is a unit vector?

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A vector having a magnitude of exactly one, used only to indicate direction. Example: î, ĵ, k̂ along the x, y, z axes.

1 markQ6. What is meant by a null (zero) vector?

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A vector whose magnitude is zero and whose direction is indeterminate. Example: the resultant of two equal and opposite vectors.

1 markQ7. Can the magnitude of a vector be negative?

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No. The magnitude of a vector is always a non-negative real number; a negative sign only reverses its direction.

1 markQ8. State whether displacement is a scalar or a vector.

Show answer

Displacement is a vector quantity — it has both magnitude (shortest distance) and direction (from initial to final position).

1 markQ9. Two vectors are said to be equal when — complete the statement.

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…when they have the same magnitude and the same direction, regardless of their initial points (position).

1 markQ10. What is the angle between two vectors for their resultant to be maximum?

Show answer

0° (vectors acting in the same direction); the resultant magnitude is then A + B.

Section B

Short Answer Questions

2–3 marks each

2 marksQ1. Distinguish between scalar and vector quantities with one example of each.

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Answer A scalar has magnitude and unit only, and adds by ordinary algebra (e.g., mass: 2 kg + 3 kg = 5 kg). A vector has magnitude, unit, and direction, and adds by the triangle/parallelogram law (e.g., velocity: two velocities at an angle combine geometrically, not by simple addition).

2 marksQ2. Why is electric current not considered a vector quantity even though it has direction?

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Answer Current has magnitude and a sense of direction along a wire, but two currents meeting at a junction combine algebraically (Kirchhoff's current law), not by the triangle law. Since a valid vector must obey vector addition rules, current fails this test and is classified as a scalar.

3 marksQ3. State the triangle law of vector addition and mention one limitation of representing vectors only graphically.

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Answer Triangle law: if two vectors are represented in magnitude and direction by two sides of a triangle taken in order, their resultant is represented by the third side taken in the reverse order (tail of first to head of second).

Limitation: a purely graphical (scale-drawing) method is time-consuming and gives limited accuracy compared to the analytical formula R = √(A² + B² + 2AB cosθ), especially for angles that are not simple values.

2 marksQ4. What are equal and negative vectors? Give an example of each.

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Answer Equal vectors: same magnitude and same direction (e.g., two cars moving at 40 km/h due north). Negative vectors: same magnitude but opposite direction (e.g., $\vec{A}$ and $-\vec{A}$ — a vector and its reverse).

3 marksQ5. Explain resolution of a vector into rectangular components with a labelled reasoning (no diagram needed, describe it).

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Answer Any vector $\vec{A}$ in the xy-plane can be broken into two mutually perpendicular components: $A_x = A\cos\theta$ along the x-axis and $A_y = A\sin\theta$ along the y-axis, where θ is the angle the vector makes with the x-axis. These components, added vectorially, reproduce the original vector: $\vec{A} = A_x\hat{i} + A_y\hat{j}$. This makes vector algebra (addition/subtraction) far simpler because components along the same axis just add algebraically.

2 marksQ6. Two forces of 3 N and 4 N act on a body at right angles to each other. Find the magnitude of the resultant.

Show answer

Answer Since θ = 90°, R = √(3² + 4²) = √25 = 5 N, directed at tan⁻¹(4/3) = 53° from the 3 N force.

Section C

Long Answer Questions

5 marks each — full derivations expected in exam

5 marksQ1. State and derive the expression for the magnitude and direction of the resultant of two vectors using the parallelogram law of vector addition.

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Answer (outline) Let $\vec{A}$ and $\vec{B}$ act at angle θ, represented as two adjacent sides OP and OQ of a parallelogram OPRQ from a common point O. The diagonal OR represents the resultant $\vec{R}$.

Drop a perpendicular from R to the extended OP, meeting it at N. In right triangle ONR: ON = A + B cosθ, and NR = B sinθ.

By Pythagoras: R² = (A + Bcosθ)² + (Bsinθ)² ⇒ R = √(A² + B² + 2AB cosθ).

Direction: tanα = NR / ON = B sinθ / (A + B cosθ), where α is the angle the resultant makes with $\vec{A}$.

Special cases: θ=0° gives R=A+B (maximum); θ=180° gives R=|A−B| (minimum); θ=90° gives R=√(A²+B²).

5 marksQ2. Explain the resolution of a vector in a plane into two mutually perpendicular components, and use it to derive the formula for the resultant of two vectors by the component method.

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Answer (outline) A vector $\vec{A}$ making angle θ with the x-axis has components $A_x = A\cos\theta$, $A_y = A\sin\theta$, so $\vec{A} = A_x\hat{i} + A_y\hat{j}$, and $A = \sqrt{A_x^2+A_y^2}$.

For two vectors $\vec{A} = A_x\hat{i}+A_y\hat{j}$ and $\vec{B}=B_x\hat{i}+B_y\hat{j}$, the resultant is found by adding components along each axis separately:
$R_x = A_x + B_x$, $R_y = A_y + B_y$
$\vec{R} = R_x\hat{i} + R_y\hat{j}$, with magnitude $R = \sqrt{R_x^2 + R_y^2}$ and direction $\theta = \tan^{-1}(R_y/R_x)$.

This component method avoids drawing diagrams for every problem and is the standard technique used in numericals involving 3 or more vectors.

5 marksQ3. Distinguish between scalar (dot) product and vector (cross) product of two vectors, giving their definitions, formulae, and one physical example of each.

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Answer Scalar (dot) product: $\vec{A}\cdot\vec{B} = AB\cos\theta$, a scalar result. It represents the component of one vector along another. Physical example: Work done, $W = \vec{F}\cdot\vec{d} = Fd\cos\theta$.

Vector (cross) product: $\vec{A}\times\vec{B} = AB\sin\theta\,\hat{n}$, a vector result, where $\hat{n}$ is perpendicular to the plane of $\vec{A}$ and $\vec{B}$ (direction by the right-hand rule). Physical example: Torque, $\vec{\tau} = \vec{r}\times\vec{F}$.

Key differences: dot product is commutative ($\vec{A}\cdot\vec{B}=\vec{B}\cdot\vec{A}$), cross product is anti-commutative ($\vec{A}\times\vec{B}=-\vec{B}\times\vec{A}$); dot product is maximum when vectors are parallel (θ=0°), cross product is maximum when perpendicular (θ=90°).

Section D

Multiple Choice Questions

1 mark each — correct option highlighted in the reveal

Q1. Which of the following is a scalar quantity?

• A. Momentum
• B. Electric current
• C. Force
• D. Displacement

Show answer

Correct option: B. Electric current — has direction but does not obey the triangle law, so it is a scalar.

Q2. The resultant of two vectors of magnitude 3 and 4 units acting at 90° to each other is:

• A. 1 unit
• B. 7 units
• C. 5 units
• D. 25 units

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Correct option: C. 5 units — √(3²+4²) = √25 = 5.

Q3. Two vectors are equal if they have:

• A. the same magnitude only
• B. the same direction only
• C. the same magnitude and direction
• D. the same initial point

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Correct option: C. the same magnitude and direction.

Q4. The maximum number of components a vector can be resolved into is:

• A. Exactly two
• B. Exactly three
• C. Any number, but two mutually perpendicular components are most commonly used
• D. Only one

Show answer

Correct option: C — a vector can be resolved into any number of components, but resolving into two (or three, in 3D) mutually perpendicular components is standard practice.

Q5. If $\vec{A} + \vec{B} = \vec{A} - \vec{B}$, then:

• A. $\vec{A} = 0$
• B. $\vec{B} = 0$
• C. Both are zero
• D. $\vec{A} = \vec{B}$

Show answer

Correct option: B. $\vec{B}=0$ — the equation simplifies to 2$\vec{B}$ = 0.

Q6. A unit vector has:

• A. Magnitude 1 and no unit
• B. Magnitude equal to the vector it represents direction for
• C. Zero magnitude
• D. Magnitude 10

Show answer

Correct option: A. Magnitude 1 and no unit — it is dimensionless and purely indicates direction.

Q7. The dot product of two mutually perpendicular vectors is:

• A. Maximum
• B. Equal to AB
• C. Zero
• D. Negative

Show answer

Correct option: C. Zero — since cos 90° = 0.

Section E

Assertion & Reason

Each question has an Assertion (A) and a Reason (R). Choose the correct option:

• (a) Both A and R are true, and R is the correct explanation of A
• (b) Both A and R are true, but R is NOT the correct explanation of A
• (c) A is true, R is false
• (d) A is false, R is true

Q1. Assertion (A): Electric current is not a vector quantity.
Reason (R): Electric current does not obey the triangle law of vector addition.

Show answer

Correct option: (a) — both true, and R correctly explains A; current has direction but fails the addition test required of vectors.

Q2. Assertion (A): The magnitude of the resultant of two vectors can never be less than the difference of their magnitudes.
Reason (R): The resultant is minimum when the two vectors act in the same direction.

Show answer

Correct option: (c) — A is true (minimum resultant = |A−B|), but R is false: the resultant is minimum when vectors act in opposite directions (θ = 180°), not the same direction.

Q3. Assertion (A): Two vectors of unequal magnitude can never give a zero resultant.
Reason (R): A zero resultant requires the two vectors to be exactly equal in magnitude and opposite in direction.

Show answer

Correct option: (a) — both true and R correctly explains A. Only two vectors of equal magnitude acting in exactly opposite directions can cancel to give a null vector.

Q4. Assertion (A): A physical quantity having both magnitude and direction is always a vector.
Reason (R): Electric current has both magnitude and direction, yet it is a scalar.

Show answer

Correct option: (d) — A is false (having magnitude and direction alone doesn't guarantee vector status; it must also obey the triangle law); R is a true, independent statement that in fact contradicts A.

Q5. Assertion (A): The scalar (dot) product of two vectors can be negative.
Reason (R): cos θ is negative for angles between 90° and 180°.

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Correct option: (a) — both true, and R correctly explains why A holds.

Section F

Fill in the Blanks

Q1. A quantity having magnitude only and no direction is called a .

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scalar

Q2. The rear end of a vector arrow is called the , and the front end is called the .

Show answer

tail; head

Q3. Two vectors acting in exactly opposite directions are called vectors.

Show answer

negative

Q4. The resultant of two vectors is maximum when the angle between them is .

Show answer

0°

Q5. The resultant of two vectors is minimum when the angle between them is .

Show answer

180°

Q6. A vector whose magnitude is zero is called a vector.

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null (zero)

Q7. The dot product of two vectors is also known as the product.

Show answer

scalar

Q8. The cross product of two vectors is also known as the product, and its result is always a .

Show answer

vector; vector (perpendicular to the plane of the two vectors)

Q9. $\hat{i}, \hat{j}, \hat{k}$ are examples of vectors along the x, y, z axes.

Show answer

unit

Q10. The law used to find the resultant of two vectors represented as adjacent sides of a figure from a common point is called the law.

Show answer

parallelogram

Section G

Match the Column

Match Column A (quantity) with Column B (type):

Column A	Column B
1. Mass	(p) Vector
2. Displacement	(q) Scalar
3. Electric current	(r) Vector
4. Force	(s) Scalar

Show answer

1 → (q) Scalar  |  2 → (p) Vector  |  3 → (s) Scalar  |  4 → (r) Vector

Match the angle between two vectors (Column A) with the type of resultant (Column B):

Column A	Column B
1. θ = 0°	(p) R = |A − B| (minimum)
2. θ = 90°	(q) R = A + B (maximum)
3. θ = 180°	(r) R = √(A² + B²)

Show answer

1 → (q)  |  2 → (r)  |  3 → (p)

Section H

Statement-Based Questions

Read the statements and choose: (a) Both true, (b) Statement I true, II false, (c) Statement I false, II true, (d) Both false

Q1.

Statement I: Every vector has both magnitude and direction.
Statement II: Every quantity with magnitude and direction is a vector.

Show answer

Correct option: (b) — Statement I is true by definition. Statement II is false, since current disproves it (it fails the triangle law).

Q2.

Statement I: The scalar product of two perpendicular vectors is zero.
Statement II: The vector product of two parallel vectors is zero.

Show answer

Correct option: (a) — Both true. Dot product ∝ cosθ = 0 at 90°; cross product ∝ sinθ = 0 at 0°/180° (parallel).

Q3.

Statement I: Vector addition is commutative.
Statement II: Vector subtraction is commutative.

Show answer

Correct option: (b) — $\vec{A}+\vec{B}=\vec{B}+\vec{A}$ is true, but $\vec{A}-\vec{B} \neq \vec{B}-\vec{A}$ in general, so Statement II is false.

Section I

Case Study Based Question

A student is studying a small ball moving inside a long straight tube. While the ball moves along the length of the tube at a steady speed, the tube itself is being carried across the room, moving in a direction perpendicular to its own length, at a different steady speed. The student wants to determine the actual velocity of the ball as observed by someone standing still in the room (not moving with the tube).

(i) Which law of vector addition should the student use to find the ball's resultant velocity?

Show answer

The triangle law of vector addition (equivalently, the parallelogram law), since the two velocities act at an angle to each other, not along the same line.

(ii) If the tube moves at 3 m/s and the ball moves at 4 m/s relative to the tube, find the magnitude of the resultant velocity.

Show answer

Since the two velocities are perpendicular: R = √(3² + 4²) = √25 = 5 m/s.

(iii) Find the angle the resultant velocity makes with the direction of the tube's motion.

Show answer

θ = tan⁻¹(4/3) = 53° from the direction of the tube's velocity.

(iv) If instead the ball's velocity along the tube were reversed in sense (but same magnitude), would the magnitude of the resultant velocity change?

Show answer

No. Reversing one component's sense changes the resultant's direction, but since the two velocities remain perpendicular with the same magnitudes (3 and 4), the resultant magnitude stays 5 m/s.

CBSE CLASS 11 PHYSICS · VECTORS & SCALARS · COMPLETE QUESTION BANK