To add two vectors A and B graphically, place B's tail at A's head; the sum R = A + B is the vector from A's tail to B's head. Equivalently, when A and B emanate from a common origin and form two adjacent sides of a parallelogram, the diagonal from that origin is R.
-- NCERT Class 11 Physics, Ch. 3, p. 3Vector Addition Subtraction
Lesson
Vector addition and subtraction is about combining displacement-like, velocity-like, or force-like quantities that carry both magnitude and direction. The operation is NOT ordinary arithmetic — 3 km east plus 4 km north does NOT give 7 km.
Triangle law (NCERT Class 11 Physics Chapter 3, page 3). Place the tail of the second vector at the head of the first. The resultant runs from the tail of the first to the head of the second. Order does not matter: A + B = B + A (commutative).
Parallelogram law (same reference). Place both vectors tail-to-tail. Complete the parallelogram. The diagonal from the common tail is the resultant. The magnitude formula (NCERT Class 11 Physics Chapter 3, page 8):
R = √(A² + B² + 2AB cos θ)
where θ is the angle between A and B when placed tail-to-tail.
Direction of the resultant. The angle α that R makes with A is:
tan α = B sin θ / (A + B cos θ)
Special cases worth memorising:
- θ = 0° (parallel): R = A + B (maximum).
- θ = 180° (anti-parallel): R = |A − B| (minimum).
- θ = 90°: R = √(A² + B²).
Vector subtraction. A − B = A + (−B). Reverse B, then add using the triangle or parallelogram law. The magnitude of A − B uses the same formula with θ replaced by (180° − θ): R_sub = √(A² + B² − 2AB cos θ).
Common confusion in NEET: mixing up the sign of the 2AB cos θ term between addition and subtraction. Addition uses +2AB cos θ; subtraction uses −2AB cos θ (equivalently, cos(180° − θ) = −cos θ). A second frequent error is forgetting that the angle θ must be measured when the vectors are placed tail-to-tail, not head-to-tail.
Practice MCQs
Select an option to see the explanation. Wrong answers show why your choice was tempting — and name the exact trap it exploits.
Two vectors of magnitudes 3 units and 4 units are added. Which of the following CANNOT be the magnitude of their resultant?
Two forces of equal magnitude F act at an angle of 120° to each other. The magnitude of their resultant is:
If |**A** + **B**| = |**A** − **B**|, what is the angle between **A** and **B**?
Two vectors **A** and **B** have magnitudes 6 and 8 units. Their resultant has magnitude 10 units. The angle between **A** and **B** is:
The resultant of two vectors **A** and **B** is perpendicular to **A**. If |**A**| = 3 and |**B**| = 5, the magnitude of the resultant is:
If **A** = 3î + 4ĵ and **B** = −3î + 4ĵ, the magnitude of **A** − **B** is:
Two vectors of magnitudes A and B (A > B) are inclined at angle θ. The angle α that the resultant makes with vector **A** is given by:
Two equal forces F act on a body. If the resultant force is also equal to F, the angle between the two forces is:
Quick recall before you leave
Worked Example
- 1
Given
- A = 12 N, B = 5 N, θ = 60°
- 2
Required
- (a) Magnitude of resultant R - (b) Angle α between R and A (the 12 N force)
- 3
Concept
Parallelogram law of vector addition. Two vectors placed tail-to-tail define a parallelogram; the diagonal gives the resultant.
- 4
Formula
- R = √(A² + B² + 2AB cos θ) - tan α = B sin θ / (A + B cos θ)
- 5
Substitution
- R = √(12² + 5² + 2 × 12 × 5 × cos 60°) - R = √(144 + 25 + 120 × 0.5) - R = √(144 + 25 + 60)
- 6
Calculation
- R = √229 ≈ 15.13 N - tan α = 5 × sin 60° / (12 + 5 × cos 60°) - tan α = 5 × (√3/2) / (12 + 2.5) - tan α = (5√3/2) / 14.5 - tan α = 4.330 / 14.5 ≈ 0.2986 - α = arctan(0.2986) ≈ 16.6° **Note on exact values:** The angle 60° is exact (a standard angle), and the integers 5 and 12 are exact given values. These do not limit significant figures; the precision of the answer is set by the calculation.
- 7
Final answer
- (a) R ≈ 15.1 N - (b) α ≈ 16.6° from the 12 N force
- 8
Common trap
Using cos θ with the wrong sign. If you accidentally compute R = √(A² + B² − 2AB cos 60°), you get √(169 − 60) = √109 ≈ 10.4 N — that's the magnitude of **A** − **B**, not **A** + **B**. The subtraction formula uses −2AB cos θ; the addition formula uses +2AB cos θ.
- 9
Similar NEET-style question
Two displacement vectors of magnitudes 7 m and 24 m are inclined at 90° to each other. Find the magnitude and direction of the resultant displacement. (Answer: R = 25 m, α = arctan(24/7) ≈ 73.7° from the 7 m vector — a 7-24-25 Pythagorean triplet.) ---
Before solving, remember these
For vectors A and B making angle θ between them, the magnitude of R = A + B is R = √(A² + B² + 2 A B cos θ). The angle α that R makes with A satisfies tan α = (B sin θ) / (A + B cos θ).
-- NCERT Class 11 Physics, Ch. 3, p. 8Example 3.3 — Resultant of two velocities
A motorboat travels north at 25 km/h while a current flows 60° east of south at 10 km/h. Using the law of cosines (angle between vectors = 120°), the resultant speed is R ≈ √(25² + 10² + 2·25·10·(-1/2)) ≈ 22 km/h.
-- NCERT Class 11 Physics, Ch. 3, p. 8Formulas
6 formulas — click to collapse
First kinematic equation (uniform acceleration)
Final velocity equals initial velocity plus acceleration times the time elapsed, for motion under constant acceleration.
| Symbol | Quantity | SI Unit |
|---|---|---|
| v | Final velocity | m/s |
| v0 | Initial velocity | m/s |
| a | Constant (uniform) acceleration | m/s^2 |
| t | Elapsed time | s |
Valid when
- Acceleration a is CONSTANT (uniform) in both magnitude and direction
- All quantities measured in the same inertial reference frame
- Motion is along a straight line; signs encode direction along chosen axis
Do NOT use when
- Acceleration changes in magnitude or direction (use a(t) integration)
- Motion is uniformly circular at constant speed (a is centripetal, not tangential)
Second kinematic equation (displacement under uniform acceleration)
Displacement equals initial-velocity-times-time plus half of acceleration-times-time-squared. The (1/2) factor is the area of the triangle on the v-t graph.
| Symbol | Quantity | SI Unit |
|---|---|---|
| x | Final position | m |
| x0 | Initial position | m |
| v0 | Initial velocity | m/s |
| a | Constant acceleration | m/s^2 |
| t | Time elapsed | s |
Valid when
- Acceleration constant (magnitude and direction)
- Sign convention consistent across x, v, a (one chosen positive direction)
Third kinematic equation (velocity-squared)
Relates final velocity to initial velocity, displacement, and acceleration without using time. Most useful when t is unknown or unwanted.
| Symbol | Quantity | SI Unit |
|---|---|---|
| v | Final velocity | m/s |
| v0 | Initial velocity | m/s |
| a | Constant acceleration | m/s^2 |
| x - x0 | Displacement | m |
Valid when
- Constant acceleration
- Use signed values for v, v0, a, and (x - x0) consistently
Do NOT use when
- Time-dependent acceleration
- Curvilinear motion where acceleration is not parallel to displacement
Projectile maximum height
Maximum height attained by a projectile launched at speed v0 and angle theta0 above the horizontal, measured above the launch level.
| Symbol | Quantity | SI Unit |
|---|---|---|
| H | Maximum height (above launch) | m |
| v0 | Launch speed | m/s |
| theta0 | Launch angle | rad/deg |
| g | Gravitational acceleration | m/s^2 |
Valid when
- Air resistance neglected
- Constant g over trajectory
Projectile horizontal range
For a projectile launched from and returning to the same horizontal level with initial speed v0 at angle theta0 above the horizontal, the horizontal range R is given by this formula. R is maximised at theta0 = 45 deg.
| Symbol | Quantity | SI Unit |
|---|---|---|
| R | Horizontal range | m |
| v0 | Launch speed | m/s |
| theta0 | Launch angle above horizontal | rad (or deg with sin in deg) |
| g | Gravitational acceleration | m/s^2 |
Valid when
- Launch and landing are at the same vertical height
- Air resistance neglected
- g treated as constant over the trajectory
Do NOT use when
- Launch and landing heights differ (use full kinematics)
- Significant air drag (e.g. table-tennis ball, badminton shuttle)
- Variation of g (ballistic trajectories spanning large altitude changes)
Centripetal acceleration in uniform circular motion
An object moving in a circle of radius r at constant speed v has acceleration of magnitude v^2/r (or equivalently omega^2 * r) directed toward the centre. This is centripetal (radially inward), not tangential.
| Symbol | Quantity | SI Unit |
|---|---|---|
| a_c | Centripetal acceleration | m/s^2 |
| v | Tangential speed | m/s |
| r | Radius of circle | m |
| omega | Angular speed | rad/s |
Valid when
- Speed v is constant (uniform circular motion)
- r and the centre are well-defined (instantaneous radius of curvature for general curved motion)
Do NOT use when
- Non-uniform circular motion (then there is also a tangential acceleration component)
Exam Traps & Common Mistakes
These are the exact patterns that cause wrong answers in NEET. Each trap includes when it triggers and how to avoid it.
14 items — click to collapse
Category: Similar Terms
Student answers 1:2:3:4 for distances in successive 1-second intervals (linear) instead of 1:3:5:7 (Galileo's odd numbers).
When it triggers
Question asks about ratios of distances traversed in successive 1-s intervals during free fall from rest.
How to avoid
Distance grows quadratically (y = ½ g t²); successive interval distances are y_n - y_{n-1} = ½ g (t_n² - t_{n-1}²) = ½ g (2n-1) seconds. The factor (2n-1) gives 1, 3, 5, 7, ...
Category: Graph Interpretation
Student uses sin or cos of the angle the line makes with the time axis, instead of tan, to extract velocity.
When it triggers
Question gives an angle the x-t line makes with the t-axis (often 30°, 45°, 60°) and asks for velocity or its ratio.
How to avoid
Velocity = dx/dt = slope of x-t line = tan(angle), where the angle is measured from the time axis. Always tan, not sin or cos.
Category: Overthinking
Student attempts to invert t(x) algebraically before differentiating, getting tangled in messy algebra; misses chain rule.
When it triggers
Question gives t as function of x (instead of x as function of t), e.g. t = x² + x.
How to avoid
Differentiate the given relation directly: dt/dx = (function of x). Then v = dx/dt = 1/(dt/dx). For acceleration use chain rule: a = dv/dt = (dv/dx)(dx/dt) = v dv/dx.
Category: Sign Convention
Student treats a 'thrown vertically downward' problem as if the object were dropped (u = 0). The result is wrong by an additive u² term in v² = u² + 2gh. When the question explicitly states a launch speed, that speed is non-zero and CANNOT be ignored.
When it triggers
Question phrases: 'thrown vertically downward', 'projected with initial velocity', 'launched with speed u'. If u is given numerically, it MUST appear in the equation.
How to avoid
Always parse the launch verbal cue and write down u with its sign before reaching for v² = 2gh. Use the full v² = u² + 2gh (or u² - 2gh for upward motion).
Category: Similar Terms
Student plugs into v₀² sin(2θ)/g (range) when asked for maximum height, or vice versa. The two share v₀ and θ but have different sin-vs-sin² and 2g-vs-g terms.
When it triggers
Question mentions launch speed and angle and asks for max height (H) or range (R). Distractors include the wrong formula's answer.
How to avoid
Memorise BOTH formulas explicitly: H = v₀² sin² θ / (2g) (note sin²); R = v₀² sin(2θ) / g (note sin of doubled angle). Check by setting θ = 45°: max range, half max height.
Category: Sign Convention
Student plugs angle θ into v cos θ when the question states 'angle with the vertical' (which makes the horizontal component v sin θ).
When it triggers
Question phrases like 'thrown at angle θ with the vertical direction' or 'with horizontal'.
How to avoid
Always identify reference axis explicitly. From horizontal: vx = v cos θ, vy = v sin θ. From vertical: vx = v sin θ, vy = v cos θ. The two are complementary (θ_h + θ_v = 90°).
Category: Sign Convention
Student fails to distinguish between same-direction and opposite-direction relative velocities, treating both as magnitudes.
When it triggers
Question describes two objects moving on the same line; observer somewhere between or alongside.
How to avoid
Relative velocity is a VECTOR. Same direction: v_rel = v_a - v_b (smaller magnitude). Opposite direction: v_rel = v_a + v_b (larger magnitude). Use sign convention consistently along chosen axis.
Category: Overthinking
Student assumes proportionality of speed to remaining distance under uniform deceleration. In fact, KE drops linearly with distance (v² is the linear quantity, not v): v² = u² - 2as. Speed-vs-distance is a sqrt-curve, not a line.
When it triggers
Question describes a body decelerating through stages with given speed at one stage; asks for distance to stop or speed at another stage.
How to avoid
Always work with v², not v, when uniform deceleration is in play. The work-energy theorem gives the same answer faster: ½ m v² = work done against constant force over distance.
Category: Overthinking
Student uses the radius R as the projectile launch height or fails to compute the UCM speed from period.
When it triggers
Question describes a particle in UCM with given (R, T) then says 'now launched vertically up with same speed; find max height'.
How to avoid
Step 1: speed v = 2πR/T (from UCM). Step 2: max projectile height H = v²/(2g) = (2πR/T)² / (2g). Don't shortcut by setting H = R.
Category: Similar Terms
Student claims velocity is constant in uniform circular motion (it's not — direction changes).
When it triggers
Question asks 'in uniform circular motion at constant speed, which is also constant?'
How to avoid
In UCM: SPEED constant; KE constant. VELOCITY (vector) NOT constant. ACCELERATION (centripetal, magnitude v²/r) constant in MAGNITUDE but NOT in direction.
Root cause: concept gap
Correction
Average velocity = total displacement / total time (vector). Average speed = total path length / total time (scalar, always >= |average velocity|). For round-trip motion, average velocity is zero; average speed is not.
Wrong option pattern
Distractor offers (v1+v2)/2 instead of total-distance/total-time.
Root cause: formula misuse
Correction
The three kinematic equations require CONSTANT acceleration. For variable acceleration, use a = dv/dt and integrate, or use v dv = a dx for position-dependent acceleration. Verify constant-a before applying these formulas.
Wrong option pattern
Distractor uses constant-acceleration kinematic equations on a problem where the question explicitly says acceleration changes with time or position.
Root cause: concept gap
Correction
Standard range R = v0^2 sin(2*theta)/g and H = v0^2 sin^2(theta)/(2g) assume (i) launch and landing at the same height, (ii) negligible air drag, and (iii) constant g. For asymmetric trajectories, use the full kinematic decomposition along x and y.
Wrong option pattern
Distractor applies R = v0^2 sin(2*theta)/g to a projectile launched from a cliff.
Root cause: concept gap
Correction
Acceleration is the rate of change of VELOCITY (a vector), not speed. In uniform circular motion, the speed is constant but the velocity direction changes continuously, giving a centripetal acceleration of magnitude v^2/r toward the centre.
Wrong option pattern
Distractor option says 'a = 0 because speed is constant'.
Past Year Questions
10 questions from NEET 2020, 2021, 2022, 2023, 2024, 2025. Answers verified against NTA official keys. — click to collapse
A particle moving with uniform speed in a circular path maintains:
The ratio of the distances travelled by a freely falling body in the 1st, 2nd, 3rd and 4th second
How NEET usually asks this
10 recurring patterns from past papers — click to collapse
For an object dropped from rest, distances traversed in successive equal time intervals are in the ratio 1:3:5:7:9:... (odd numbers). Derivable from y = ½ g t². Common shape: 'find ratio of distance covered in 1st, 2nd, 3rd, 4th seconds of free fall'.
Common distractors
uses arithmetic progression 1 2 3 4
Linear-time intuition
Time given as implicit function of position, e.g. t = x² + x. To find acceleration, differentiate twice: dt/dx = 2x + 1, so v = dx/dt = 1/(2x+1); a = dv/dt = -2v²·v = -2v³ (chain rule). Multi-step calculus.
Common distractors
treats t x as explicit
Default to t-as-input thinking
A projectile (typically a bullet) penetrates a uniform medium with constant decelerating force; given initial speed and speed after a known distance, find the total stopping distance. Apply v² = u² + 2as to each segment, noting that 'a' is the same throughout. Common shape: bullet hits block at u, slows to u/k after distance d₁; how much further to stop?
Common distractors
treats speed ratio as distance ratio
Linear thinking: 'speed went from u to u/3, so distance traversed should be 3× the original'
Two particles' position-time graphs are straight lines with given angles vs the time axis (e.g. 30° and 45°); find ratio of velocities. Velocity = slope = tan(angle). Common shape: ratio = tan(30°)/tan(45°) = 1/√3.
Common distractors
uses sin or cos instead of tan
Trig confusion under time pressure
A particle in uniform circular motion (period T, radius R) is subsequently launched vertically upward with the SAME speed (v = 2πR/T); find max height. H = v²/(2g) = (2πR/T)²/(2g).
Common distractors
uses radius as height
Surface confusion of geometric R and projectile H
Given launch speed v₀ and angle θ above horizontal, find maximum height. H = v₀² sin² θ / (2g). Common values plug in cleanly when sin θ ∈ {0.5, 0.707, 0.866}. Distractors test (i) using sin instead of sin², (ii) forgetting the factor of 2 in denominator, (iii) using g = 10 vs g = 9.8 inconsistently.
Common distractors
uses sin instead of sin squared
Confusing the height formula H = v²sin²θ/(2g) with the range 2v sin θ / g
Projectile launched at angle θ; find SPEED at highest point. The vertical component vanishes at the apex; horizontal component v₀ cos θ is conserved. Common gotcha: 'angle with the vertical' vs 'angle with the horizontal'.
Common distractors
uses sin instead of cos
Sign-of-angle-axis confusion
speed zero at top
True only for vertical motion (no horizontal v₀)
Buses leave both ends of a route every T minutes; observer travels between them at constant speed; given separate periods at which buses pass observer in same and opposite directions, find bus speed or T. Use relative-velocity equations: same direction T_same = L / (v_bus - v_obs), opposite T_opp = L / (v_bus + v_obs).
Common distractors
ignores direction relative to observer
Treating bus-passing time as absolute
Particle in uniform circular motion at constant speed; identify which of speed / velocity / acceleration / kinetic energy is constant. Speed and KE constant; velocity (vector) and acceleration (centripetal) are NOT constant — direction changes.
Common distractors
treats velocity as scalar
Conflates speed (scalar) with velocity (vector)
Object given a non-zero downward initial velocity from elevation; asked for the height fallen, time of impact, or final speed. Apply v² = u² + 2gh (or analogous) with proper signs. Common shape: a ball thrown vertically downward from a tower with initial speed u, hitting the ground at speed v; find the tower height. Distractors test (i) sign-of-u confusion, (ii) using v² = 2gh forgetting u², (iii) wrong g unit.
Common distractors
drops initial velocity term
Student conflates 'thrown' with 'dropped' and uses v² = 2gh
Sources
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