Resolution of a vector means expressing a single vector as the sum of two or more component vectors along chosen directions. The most common choice is a pair of mutually perpendicular axes — the rectangular components (NCERT Class 11 Physics Chapter 3, page 32).
Given a vector A making angle θ with the positive x-axis:
The original vector is recovered by A = Aₓ x̂ + Aᵧ ŷ, with magnitude A = √(Aₓ² + Aᵧ²) and direction tan θ = Aᵧ / Aₓ.
Where aspirants lose marks:
Angle-reference confusion. If the angle is given from the y-axis (or "with the vertical"), the components swap: the x-component becomes A sin θ and the y-component becomes A cos θ. Always identify the reference axis before writing cos or sin.
Sign errors in quadrants. When the vector lies in the second quadrant (90° < θ < 180°), cos θ is negative — so Aₓ is negative. Students who plug in the acute angle without thinking about sign get the wrong component.
Confusing resolution with addition. Resolution decomposes ONE vector into components; vector addition combines TWO or more vectors into a resultant. The formulas look similar but serve opposite purposes.
Non-perpendicular resolution. Components along non-orthogonal axes exist and require the sine rule or parallelogram law, but NEET questions overwhelmingly use rectangular axes. If the problem doesn't specify oblique axes, default to rectangular.
Resolution is the foundation for projectile motion (separate x and y kinematics) and inclined-plane dynamics (parallel and perpendicular to slope). Master the angle-reference check here, and those topics become mechanical.
Select an option to see the explanation. Wrong answers show why your choice was tempting — and name the exact trap it exploits.
A force of 10 N acts at 60° above the positive x-axis. What is the x-component of this force?
Answer: C. The x-component is F cos θ = 10 cos 60° = 10 × 0.5 = 5 N (NCERT Class 11 Physics Chapter 3, page 32).
Why A is wrong: A is wrong because 10 N is the magnitude of the full vector, not its x-component. Resolution always yields a component smaller than or equal to the original magnitude (equality only at 0° or 90° for the respective axis).
Why B is wrong: B is wrong because 5√3 ≈ 8.66 N is the y-component (F sin 60° = 10 × √3/2), not the x-component. This results from swapping sin and cos.
Why D is wrong: D is wrong because 10√3 ≈ 17.3 N exceeds the original 10 N magnitude. A rectangular component cannot exceed the vector's magnitude.
A velocity vector of magnitude 20 m/s makes an angle of 30° with the vertical. What is its horizontal component?
Answer: C. The angle is measured from the vertical (y-axis), so the horizontal (x) component is 20 sin 30° = 10 m/s. When angle is from the vertical, horizontal = magnitude × sin θ (NCERT Class 11 Physics Chapter 3, page 32).
Why A is wrong: A is wrong because cos 30° gives the component along the axis from which the angle is measured — here that is the vertical axis, not the horizontal. 20 cos 30° = 10√3 m/s would be the vertical component.
Why B is wrong: B is wrong because tan has no role in rectangular resolution. Components are always magnitude × cos or magnitude × sin of the included angle. Tan appears in finding the angle from known components (tan θ = Aᵧ/Aₓ), not in decomposing.
Why D is wrong: D is wrong because dividing by cos θ would give a value larger than the original magnitude (20/cos 30° ≈ 23.1 m/s). A component can never exceed the vector's magnitude.
Which of the following is the correct statement about the rectangular components of a vector?
Answer: D. Rectangular resolution means projecting the vector onto two mutually perpendicular directions, yielding Aₓ = A cos θ and Aᵧ = A sin θ (NCERT Class 11 Physics Chapter 3, page 32).
Why A is wrong: A is wrong because for perpendicular resolution, |Aₓ| = |A cos θ| ≤ A and |Aᵧ| = |A sin θ| ≤ A. A component's magnitude never exceeds the original vector's magnitude.
Why B is wrong: B is wrong because |Aₓ| + |Aᵧ| = A(|cos θ| + |sin θ|), which equals A only when θ = 0° or 90° (one component vanishes). In general, |cos θ| + |sin θ| ≠ 1; the correct relation is Aₓ² + Aᵧ² = A² (Pythagorean).
Why C is wrong: C is wrong because components carry sign information. In the second quadrant, the x-component is negative (cos θ < 0 for 90° < θ < 180°). Signs encode direction.
A vector A has components Aₓ = −3 units and Aᵧ = 4 units. The magnitude of A is:
Answer: B. A = √(Aₓ² + Aᵧ²) = √(9 + 16) = √25 = 5 units (NCERT Class 11 Physics Chapter 3, page 32). The negative sign of Aₓ does not affect the magnitude since it is squared.
Why A is wrong: A is wrong because 1 = |Aₓ + Aᵧ| = |−3 + 4|. This adds components algebraically rather than using the Pythagorean relation. Components along perpendicular axes combine as squares, not linearly.
Why C is wrong: C is wrong because 7 = |Aₓ| + |Aᵧ| = 3 + 4. This is the sum of magnitudes of the components, not the magnitude of the resultant vector. The correct relation is A² = Aₓ² + Aᵧ², not A = |Aₓ| + |Aᵧ|.
Why D is wrong: D is wrong because 25 = Aₓ² + Aᵧ². This is A², not A. The final step — taking the square root — was skipped.
The x and y components of a vector are equal in magnitude. The angle this vector makes with the positive x-axis is:
Answer: B. If Aₓ = Aᵧ, then A cos θ = A sin θ, which gives tan θ = 1, so θ = 45° (NCERT Class 11 Physics Chapter 3, page 32).
Why A is wrong: A is wrong because at 0°, Aₓ = A and Aᵧ = 0. The components are not equal — one is maximum and the other is zero.
Why C is wrong: C is wrong because at 30°, Aₓ = A cos 30° = A√3/2 and Aᵧ = A sin 30° = A/2. These are unequal (ratio √3:1).
Why D is wrong: D is wrong because at 60°, Aₓ = A cos 60° = A/2 and Aᵧ = A sin 60° = A√3/2. The y-component is larger than the x-component (ratio 1:√3), not equal.
A vector of magnitude 50 units lies in the x-y plane. Its y-component is 25 units. The angle the vector makes with the x-axis is:
Answer: D. Aᵧ = A sin θ → 25 = 50 sin θ → sin θ = 0.5 → θ = 30° (NCERT Class 11 Physics Chapter 3, page 32).
Why A is wrong: A is wrong because sin 60° = √3/2 ≈ 0.866, giving Aᵧ = 50 × 0.866 ≈ 43.3, not 25. This results from confusing sin 30° and sin 60°.
Why B is wrong: B is wrong because sin 45° = 1/√2 ≈ 0.707, giving Aᵧ = 50 × 0.707 ≈ 35.4, not 25.
Why C is wrong: C is wrong because at 90° the entire vector is along y: Aᵧ = 50, not 25. This would also mean Aₓ = 0.
A force F is resolved into two components along directions that are NOT perpendicular to each other. Which of the following is true?
Answer: A. For non-perpendicular resolution, the parallelogram law applies. Depending on the angle between the chosen directions, one or both components can exceed the magnitude of the original vector. This is unlike rectangular resolution where each component ≤ the original magnitude (NCERT Class 11 Physics Chapter 3, page 31 resolves a vector along any two directions; that a component can exceed the vector is not stated there).
Why B is wrong: B is wrong because the constraint |component| ≤ |F| holds only for perpendicular (rectangular) resolution. For oblique axes, the parallelogram construction can produce components larger than F. Example: resolve a vector along two directions that make a small angle with each other — both components become large.
Why C is wrong: C is wrong because there is no reason the two components must be equal. Equality occurs only in the special case where the vector bisects the angle between the two resolution directions.
Why D is wrong: D is wrong because the magnitude relation for non-perpendicular resolution follows the parallelogram law (involving the angle between component directions), not simple addition. |F₁| + |F₂| = |F| has no general validity.
A displacement vector has magnitude 100 m. When resolved along two perpendicular axes, the component along one axis is 60 m. The component along the other axis is:
Answer: A. For perpendicular components: A² = Aₓ² + Aᵧ². So Aᵧ = √(100² − 60²) = √(10000 − 3600) = √6400 = 80 m (NCERT Class 11 Physics Chapter 3, page 32).
Why B is wrong: B is wrong because 40 = 100 − 60 treats the relationship as linear subtraction. Perpendicular components combine by Pythagoras (sum of squares), not by arithmetic difference.
Why C is wrong: C is wrong because 64 = 6400/100 comes from getting 100² − 60² = 6400 right and then dividing by 100 instead of taking the square root. √6400 = 80 m.
Why D is wrong: D is wrong because 160 = 100 + 60 adds the magnitudes. This violates the basic principle: the magnitude of the resultant equals √(sum of squares of components), not the sum of components.
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Given
• Magnitude of displacement: d = 200 m (exact, problem-defined)• Angle with the vertical: θ_v = 60° (exact angle)
Required
Horizontal component (dₓ) and vertical component (dᵧ).
Concept
Resolution of a vector into rectangular components. The angle is stated from the vertical (y-axis), so the component along the vertical is d cos θ_v and the component along the horizontal is d sin θ_v.
Formula
• dₓ = d sin θ_v (horizontal — perpendicular to the reference axis)• dᵧ = d cos θ_v (vertical — along the reference axis)
Substitution
• dₓ = 200 × sin 60° = 200 × (√3/2)• dᵧ = 200 × cos 60° = 200 × (1/2)
Calculation
• dₓ = 200 × 0.866 = 100√3 ≈ 173.2 m• dᵧ = 200 × 0.5 = 100 m
Note: 200 m, 60°, and the trigonometric values (sin 60° = √3/2, cos 60° = 1/2) are all exact in this problem. They do not limit significant figures.
Final answer
Horizontal component = 100√3 m ≈ 173.2 m; Vertical component = 100 m.
Common trap
If you read "60° with the vertical" but reflexively write dₓ = d cos 60° and dᵧ = d sin 60°, you swap the two answers: you'd get dₓ = 100 m and dᵧ = 173.2 m — exactly reversed. Always identify the reference axis first, then assign cos to the component along that axis and sin to the component perpendicular to it.
Similar NEET-style question
A force of 500 N acts at 30° with the vertical on a block resting on a horizontal surface. What is the horizontal component of the force pushing the block along the surface?
*(Answer: F sin 30° = 500 × 0.5 = 250 N — the horizontal component uses sin because the angle is from the vertical.)*
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A vector A in two dimensions can be expressed as A = Ax î + Ay ĵ where Ax = A cos θ and Ay = A sin θ are the rectangular components along the x and y axes; î and ĵ are unit vectors along x and y respectively. The magnitude is A = √(Ax² + Ay²); the direction tan θ = Ay/Ax.
-- NCERT Class 11 Physics, Ch. 3, p. 32More in Kinematics: 14 exam traps and mistakes · 6 formulas · 10 question patterns from its other lessons.
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