Vector Addition Subtraction

8 MCQs2 revision cards9-step worked example
Source: NCERT KinematicsPYQ coverage: NEET 2020, 2021, 2022, 2023, 2024, 2025Official key: NTA-verifiedLast reviewed: May 2026

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. AB = A + (−B). Reverse B, then add using the triangle or parallelogram law. The magnitude of AB 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.

MCQ 1Easy RecallPractice

Two vectors of magnitudes 3 units and 4 units are added. Which of the following CANNOT be the magnitude of their resultant?

MCQ 2Direct ApplicationPractice

Two forces of equal magnitude F act at an angle of 120° to each other. The magnitude of their resultant is:

MCQ 3Direct ApplicationPractice

If |**A** + **B**| = |**A** − **B**|, what is the angle between **A** and **B**?

MCQ 4Direct ApplicationPractice

Two vectors **A** and **B** have magnitudes 6 and 8 units. Their resultant has magnitude 10 units. The angle between **A** and **B** is:

MCQ 5CalculationPractice

The resultant of two vectors **A** and **B** is perpendicular to **A**. If |**A**| = 3 and |**B**| = 5, the magnitude of the resultant is:

MCQ 6Easy RecallPractice

If **A** = 3î + 4ĵ and **B** = −3î + 4ĵ, the magnitude of **A** − **B** is:

MCQ 7Easy RecallPractice

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:

MCQ 8Direct ApplicationPractice

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. 1

    Given

    - A = 12 N, B = 5 N, θ = 60°

  2. 2

    Required

    - (a) Magnitude of resultant R - (b) Angle α between R and A (the 12 N force)

  3. 3

    Concept

    Parallelogram law of vector addition. Two vectors placed tail-to-tail define a parallelogram; the diagonal gives the resultant.

  4. 4

    Formula

    - R = √(A² + B² + 2AB cos θ) - tan α = B sin θ / (A + B cos θ)

  5. 5

    Substitution

    - R = √(12² + 5² + 2 × 12 × 5 × cos 60°) - R = √(144 + 25 + 120 × 0.5) - R = √(144 + 25 + 60)

  6. 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. 7

    Final answer

    - (a) R ≈ 15.1 N - (b) α ≈ 16.6° from the 12 N force

  8. 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. 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

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. 3

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. 8

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. 8

Formulas

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.

SymbolQuantitySI Unit
vFinal velocitym/s
v0Initial velocitym/s
aConstant (uniform) accelerationm/s^2
tElapsed times

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.

SymbolQuantitySI Unit
xFinal positionm
x0Initial positionm
v0Initial velocitym/s
aConstant accelerationm/s^2
tTime elapseds

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.

SymbolQuantitySI Unit
vFinal velocitym/s
v0Initial velocitym/s
aConstant accelerationm/s^2
x - x0Displacementm

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.

SymbolQuantitySI Unit
HMaximum height (above launch)m
v0Launch speedm/s
theta0Launch anglerad/deg
gGravitational accelerationm/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.

SymbolQuantitySI Unit
RHorizontal rangem
v0Launch speedm/s
theta0Launch angle above horizontalrad (or deg with sin in deg)
gGravitational accelerationm/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.

SymbolQuantitySI Unit
a_cCentripetal accelerationm/s^2
vTangential speedm/s
rRadius of circlem
omegaAngular speedrad/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: 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.

Past Year Questions

10 questions from NEET 2020, 2021, 2022, 2023, 2024, 2025. Answers verified against NTA official keys. — click to collapse
NEET 2025

Two cities X and Y are connected by a regular bus service with a bus leaving in either direction every T min. A girl is driving scooty with a speed of 60 km/h in the direction X to Y notices that a bus goes past her every 30 minutes in the direction of her motion, and every 10 minutes in the opposite direction. Choose the correct option for the period T of the bus service and the speed (assumed constant) of the buses.

115 min, 120 km/h
29 min, 40 km/h
325 min, 100 km/h
410 min, 90 km/h
NTA Answer: Option 1(final)
NEET 2021

A particle moving in a circle of radius R with a uniform speed takes a time T to complete one revolution. If this particle were projected with the 'θ' same speed at an angle to the horizontal, the maximum height attained by it equals 4R. The θ, angle of projection, is then given by : 1 2gT 2 2 θ=sin−1 

1π2  R  1  2  2 gT θ=cos−1
2  π2 R 1 π2  2 R θ=cos−1 
32 gT  1 π2  2 R θ=sin−1 
42 gT 
NTA Answer: Option 1(final)

How NEET usually asks this

10 recurring patterns from past papers — click to collapse

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?

Multi StepMedium

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'

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.

Multi StepEasy

Common distractors

drops initial velocity term

Student conflates 'thrown' with 'dropped' and uses v² = 2gh

Sources

NCERT refs: Class 11 Physics Chapter 3, p.3 | Class 11 Physics Chapter 3, p.8

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