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. 29Vector Addition Subtraction
8 MCQs2 revision cards9-step worked example
Source: NCERT KinematicsOfficial key: NTA-verifiedLast updated: 27 Sep 2026
Vector Addition Subtraction, explained for NEET
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 29). 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 34):
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.
Can you answer these Vector Addition Subtraction 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?
Show answer and why every option is right or wrong
Answer: C. The resultant of two vectors lies between |A − B| and A + B, i.e., between |3 − 4| = 1 and 3 + 4 = 7. So the resultant can be 1, 5, or 7, but NOT 8 (NCERT Class 11 Physics Chapter 3, page 34).
Why A is wrong: A: 1 unit IS possible — it equals |A − B| when vectors are anti-parallel (θ = 180°). Choosing this suggests forgetting the minimum-resultant case.
Why B is wrong: B: 5 units IS possible — it's the resultant when vectors are perpendicular (√(9 + 16) = 5). Choosing this suggests not checking the Pythagorean case.
Why D is wrong: D: 7 units IS possible — it equals A + B when vectors are parallel (θ = 0°). Choosing this suggests forgetting the maximum-resultant case.
MCQ 2Direct ApplicationPractice
Two forces of equal magnitude F act at an angle of 120° to each other. The magnitude of their resultant is:
Show answer and why every option is right or wrong
Answer: B. R = √(F² + F² + 2F² cos 120°) = √(2F² + 2F²(−½)) = √(2F² − F²) = √(F²) = F. When two equal vectors include 120°, the resultant equals one of them in magnitude (NCERT Class 11 Physics Chapter 3, page 34).
Why A is wrong: A: 2F is the resultant when θ = 0° (parallel), not 120°. This error comes from using cos 0° = 1 instead of cos 120° = −½.
Why C is wrong: C: F√3 would result from θ = 60° (cos 60° = ½ gives R = F√3). Confusing 120° with 60° produces this distractor.
Why D is wrong: D: F/2 has no basis in the parallelogram formula for equal vectors. This likely comes from an arithmetic slip such as dividing by 2 instead of taking a square root.
MCQ 3Direct ApplicationPractice
If |A + B| = |A − B|, what is the angle between A and B?
Show answer and why every option is right or wrong
Answer: A. |A + B|² = A² + B² + 2AB cos θ. |A − B|² = A² + B² − 2AB cos θ. Setting them equal: 4AB cos θ = 0. Since A ≠ 0 and B ≠ 0, cos θ = 0, so θ = 90° (NCERT Class 11 Physics Chapter 3, page 34).
Why B is wrong: B: At θ = 60°, cos 60° = ½ ≠ 0, so the addition and subtraction magnitudes differ. This answer likely comes from guessing a 'standard' angle.
Why C is wrong: C: At θ = 0°, |A + B| = A + B and |A − B| = |A − B|. These are equal only if A or B is zero, which contradicts the premise of two non-zero vectors.
Why D is wrong: D: At θ = 180°, |A + B| = |A − B| and |A − B| = A + B. These are equal only if A = B, but the condition must hold for any magnitudes, requiring cos θ = 0.
MCQ 4CalculationPractice
Three coplanar forces act at a point: F₁ = 6 N along the +x-axis, F₂ = 8 N along the +y-axis, and F₃, such that the resultant of all three forces is a null vector. What is the magnitude of F₃?
Show answer and why every option is right or wrong
Answer: A. First find the resultant of F₁ and F₂: since they are perpendicular, R₁₂ = √(6² + 8²) = √100 = 10 N. For the resultant of all three forces to be a null vector, F₃ must be equal in magnitude and exactly opposite in direction to R₁₂ (NCERT Class 11 Physics, Chapter 3, page 30: for two vectors A and −A of equal magnitude and opposite direction, A + (−A) = 0, a null vector). So |F₃| = 10 N.
Why B is wrong: B: 2 N comes from treating F₁ and F₂ as if they were collinear and subtracting their magnitudes (|8 − 6|), ignoring that they act along perpendicular axes and must be combined by the parallelogram rule, not scalar subtraction.
Why C is wrong: C: 14 N comes from simply adding the magnitudes (6 + 8), which would only be valid if F₁ and F₂ were parallel — they are perpendicular here.
Why D is wrong: D: 7 N is the arithmetic average of 6 and 8, which has no basis in vector combination; it ignores both the perpendicular geometry and the null-vector condition.
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:
Show answer and why every option is right or wrong
Answer: D. If R is perpendicular to A, then R · A = 0. Using the component approach with R = A + B: the component of R along A must vanish, so A + B cos θ = 0 → cos θ = −A/B = −3/5. Then R = B sin θ = 5 × √(1 − 9/25) = 5 × 4/5 = 4 (NCERT Class 11 Physics Chapter 3, page 34).
Why A is wrong: A: The value 2 might come from computing |B − A| = |5 − 3| = 2 (treating vectors as scalars). Vector subtraction of magnitudes only applies when vectors are anti-parallel.
Why B is wrong: B: 8 = 3 + 5 is the resultant for parallel vectors (θ = 0°), which contradicts the perpendicularity condition on R.
Why C is wrong: C: √34 = √(9 + 25) would be R if θ = 90°, but the problem states R is perpendicular to A, not that A is perpendicular to B. These are different geometric conditions.
MCQ 6Easy RecallPractice
If A = 3î + 4ĵ and B = −3î + 4ĵ, the magnitude of A − B is:
Show answer and why every option is right or wrong
Answer: A. A − B = (3 − (−3))î + (4 − 4)ĵ = 6î + 0ĵ. Magnitude = √(36 + 0) = 6 (NCERT Class 11 Physics Chapter 3, page 34).
Why B is wrong: B: Zero would be the result of A − A, not A − B. Since B differs from A in the x-component, the difference is non-zero.
Why C is wrong: C: 8 comes from doubling the y-component (4 + 4 = 8), which is |A + B|, not |A − B|. The x-components cancel in addition; the y-components cancel in subtraction.
Why D is wrong: D: 10 = 6 + 4 or possibly √(36 + 64). Neither is correct — the y-component of the difference is zero, not 8.
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:
Show answer and why every option is right or wrong
Answer: C. From the parallelogram law, resolving B into components along and perpendicular to A: the perpendicular component is B sin θ, the along-A component is A + B cos θ. Therefore tan α = B sin θ / (A + B cos θ) (NCERT Class 11 Physics Chapter 3, page 34).
Why A is wrong: A: This swaps A and B in the formula. It would give the angle the resultant makes with B, not with A. Mixing up which vector is the reference causes this error.
Why B is wrong: B: This expression has no geometric basis in the parallelogram law. It appears to be a fabricated combination of terms.
Why D is wrong: D: This incorrectly places cos θ in the numerator and sin θ in the denominator, reversing the perpendicular and parallel components of B relative to A.
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:
Show answer and why every option is right or wrong
Answer: D. R² = F² + F² + 2F² cos θ. Setting R = F: F² = 2F² + 2F² cos θ → 2F² cos θ = −F² → cos θ = −½ → θ = 120° (NCERT Class 11 Physics Chapter 3, page 34).
Why A is wrong: A: At θ = 60°, R = F√3 ≈ 1.73F, not F. This comes from confusing cos 60° = ½ with cos 120° = −½.
Why B is wrong: B: At θ = 90°, R = F√2 ≈ 1.41F, not F. This is the perpendicular case and gives a larger resultant than F.
Why C is wrong: C: At θ = 150°, cos 150° = −√3/2, giving R = F√(2 − √3) ≈ 0.52F, which is less than F. This overshoots the required angle.
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Vector Addition Subtraction: quick recall before you leave
How do you solve a Vector Addition Subtraction question? A 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.)
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What to remember before solving Vector Addition Subtraction questions
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. 34Worked Example
Example 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. 34More in Kinematics: 14 exam traps and mistakes · 6 formulas · 10 question patterns from its other lessons.
Vector Addition Subtraction questions from past NEET papers
No question in our NEET 2020–2025 set targets this topic directly.
Sources
NCERT refs: Class 11 Physics Chapter 3, p.29 | Class 11 Physics Chapter 3, p.34
Page numbers are the ones printed in the current NCERT textbook (2023 rationalised edition), unless marked pre-2023. The books are free at ncert.nic.in.
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