- 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 |
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.
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.
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:
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.
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.
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.
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.
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.
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.
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?
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:
Since this happened in 1 s, the resultant velocity is 5 m/s, directed at
…from the direction of the tube.
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.
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.
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: $\vec{A}+\vec{B} = \vec{B}+\vec{A}$.
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 Answer Questions
1 mark each — one line / one word answers
Show answer
Show answer
Show answer
Show answer
Show answer
Show answer
Show answer
Show answer
Show answer
Show answer
Short Answer Questions
2–3 marks each
Show answer
Show answer
Show answer
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.
Show answer
Show answer
Show answer
Long Answer Questions
5 marks each — full derivations expected in exam
Show answer
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).
Show answer
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.
Show answer
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°).
Multiple Choice Questions
1 mark each — correct option highlighted in the reveal
Show answer
Show answer
Show answer
Show answer
Show answer
Show answer
Show answer
Assertion & Reason
Each question has an Assertion (A) and a Reason (R). Choose the correct option:
Reason (R): Electric current does not obey the triangle law of vector addition.
Show answer
Reason (R): The resultant is minimum when the two vectors act in the same direction.
Show answer
Reason (R): A zero resultant requires the two vectors to be exactly equal in magnitude and opposite in direction.
Show answer
Reason (R): Electric current has both magnitude and direction, yet it is a scalar.
Show answer
Reason (R): cos θ is negative for angles between 90° and 180°.
Show answer
Fill in the Blanks
Show answer
Show answer
Show answer
Show answer
Show answer
Show answer
Show answer
Show answer
Show answer
Show answer
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
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
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
Statement I: Every vector has both magnitude and direction.
Statement II: Every quantity with magnitude and direction is a vector.
Show answer
Statement I: The scalar product of two perpendicular vectors is zero.
Statement II: The vector product of two parallel vectors is zero.
Show answer
Statement I: Vector addition is commutative.
Statement II: Vector subtraction is commutative.
Show answer
Case Study Based Question
(i) Which law of vector addition should the student use to find the ball's resultant velocity?
Show answer
(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
(iii) Find the angle the resultant velocity makes with the direction of the tube's motion.
Show answer
(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
