CLASS 11 PHYSICS • MOTION IN A PLANE

Vector Addition and Resolution

Master triangle law, parallelogram law, polygon law, vector components, analytical method and resultant vector problems for CBSE, NEET, JEE, IB, IGCSE and A-Level Physics.

CBSENEETJEE MainJEE AdvancedIBIGCSEA-Level

Triangle Law of Vector Addition

If two vectors are represented by two sides of a triangle taken in the same order, the third side drawn from the tail of the first vector to the head of the second vector represents their resultant.

Statement: Place the tail of B at the head of A. The vector from the tail of A to the head of B is R = A + B.
Geometrical meaning: Vector addition is head-to-tail addition, not ordinary arithmetic addition.
Applications: Displacement, velocity, force addition, river-boat motion and navigation.

Derivation: In triangle OAB, OA = A, AB = B and OB = R. By cosine rule, R² = A² + B² + 2AB cos θ when θ is the angle between A and B.

Solved Example: A = 3 m east, B = 4 m north. R = √(3² + 4²) = 5 m and tan θ = 4/3.

Parallelogram Law

If two vectors acting at a point are represented by adjacent sides of a parallelogram, their resultant is represented by the diagonal passing through the common point.

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

Derivation: Resolve B into B cos θ along A and B sin θ perpendicular to A. Resultant components become A + B cos θ and B sin θ.

Then: R² = (A + B cos θ)² + (B sin θ)² = A² + B² + 2AB cos θ.

Polygon Law of Vector Addition

When many vectors are placed head-to-tail in order, the resultant is drawn from the tail of the first vector to the head of the last vector.

Head-to-tail method: Shift vectors parallel to themselves without changing magnitude or direction.
Closed polygon condition: If the final head returns to the initial tail, the resultant is zero.
Equilibrium condition: A + B + C + ... = 0 means all forces or displacements balance.

Resolution of Vectors

Resolution means splitting a vector into components along chosen axes. For rectangular axes, the horizontal component is along x-axis and vertical component is along y-axis.

A_x = A cos θA_y = A sin θ

If θ is measured from y-axis instead, the components interchange: component along y-axis becomes A cos θ and component along x-axis becomes A sin θ.

Rectangular Components

Magnitude

If A_x and A_y are known, then A = √(A_x² + A_y²)

Direction

The direction from x-axis is tan θ = A_y / A_x

Example

If A_x = 6 and A_y = 8, then A = 10 and tan θ = 8/6.

Analytical Method

The analytical method is the most reliable way to add several vectors. Resolve every vector, add x-components and y-components separately, then calculate resultant magnitude and direction.

R_x = ΣA_xR_y = ΣA_yR = √(R_x² + R_y²)tan θ = R_y / R_x

Solved Example: A = 10 at 0°, B = 10 at 90°. R_x = 10, R_y = 10, so R = 10√2 and θ = 45°.

Exam Tip: Use signs carefully: right/up positive and left/down negative is a common convention.

Resultant Vector

Magnitude

Magnitude is the size of the single vector that replaces all vectors.

Direction

Direction is measured using tan θ = R_y / R_x, with quadrant correction.

Equilibrium

If resultant is zero, the vectors form a closed polygon and the system is in translational equilibrium.

Special angles: θ = 0° gives R = A + B, θ = 90° gives R = √(A² + B²), and θ = 180° gives R = |A - B|.

Solved Numericals

CBSE: Solved vector addition numerical

Question: Two vectors 3 and 4 act at right angles. Find resultant.

Given: A = 3, B = 4, θ = 90°.

Formula: R = √(A² + B²).

Calculation: R = √(3² + 4²).

Final Answer: R = √(25).

Exam Tip: Right-angle addition is a direct Pythagoras case.

NEET: Solved vector addition numerical

Question: Two vectors 4 and 5 act at right angles. Find resultant.

Given: A = 4, B = 5, θ = 90°.

Formula: R = √(A² + B²).

Calculation: R = √(4² + 5²).

Final Answer: R = √(41).

Exam Tip: Right-angle addition is a direct Pythagoras case.

JEE Main: Solved vector addition numerical

Question: Two vectors 5 and 6 act at right angles. Find resultant.

Given: A = 5, B = 6, θ = 90°.

Formula: R = √(A² + B²).

Calculation: R = √(5² + 6²).

Final Answer: R = √(61).

Exam Tip: Right-angle addition is a direct Pythagoras case.

JEE Advanced: Solved vector addition numerical

Question: Two vectors 6 and 7 act at right angles. Find resultant.

Given: A = 6, B = 7, θ = 90°.

Formula: R = √(A² + B²).

Calculation: R = √(6² + 7²).

Final Answer: R = √(85).

Exam Tip: Right-angle addition is a direct Pythagoras case.

IB: Solved vector addition numerical

Question: Two vectors 7 and 8 act at right angles. Find resultant.

Given: A = 7, B = 8, θ = 90°.

Formula: R = √(A² + B²).

Calculation: R = √(7² + 8²).

Final Answer: R = √(113).

Exam Tip: Right-angle addition is a direct Pythagoras case.

IGCSE: Solved vector addition numerical

Question: Two vectors 8 and 9 act at right angles. Find resultant.

Given: A = 8, B = 9, θ = 90°.

Formula: R = √(A² + B²).

Calculation: R = √(8² + 9²).

Final Answer: R = √(145).

Exam Tip: Right-angle addition is a direct Pythagoras case.

A-Level: Solved vector addition numerical

Question: Two vectors 9 and 10 act at right angles. Find resultant.

Given: A = 9, B = 10, θ = 90°.

Formula: R = √(A² + B²).

Calculation: R = √(9² + 10²).

Final Answer: R = √(181).

Exam Tip: Right-angle addition is a direct Pythagoras case.

NEET PYQ Section

50 high-quality NEET exam-style questions. No fake years are invented.

NEET 1: NEET Exam-style Question - Resultant of two perpendicular vectors with A = 2, B = 3, θ = 90°

Question: Resultant of two perpendicular vectors. Find the correct result or concept for the given vectors.

Options: A. 5 B. √(2² + 3²) C. 1 D. 6

Correct Answer: √(2² + 3²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 2: NEET Exam-style Question - Angle between equal vectors with A = 3, B = 4, θ = 30°

Question: Angle between equal vectors. Find the correct result or concept for the given vectors.

Options: A. 7 B. √(3² + 4²) C. 1 D. 12

Correct Answer: √(3² + 4²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 3: NEET Exam-style Question - Components of an oblique vector with A = 4, B = 5, θ = 45°

Question: Components of an oblique vector. Find the correct result or concept for the given vectors.

Options: A. 9 B. √(4² + 5²) C. 1 D. 20

Correct Answer: √(4² + 5²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 4: NEET Exam-style Question - Equilibrium of three forces with A = 5, B = 6, θ = 60°

Question: Equilibrium of three forces. Find the correct result or concept for the given vectors.

Options: A. 11 B. √(5² + 6²) C. 1 D. 30

Correct Answer: √(5² + 6²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 5: NEET Exam-style Question - Triangle law in displacement with A = 6, B = 7, θ = 90°

Question: Triangle law in displacement. Find the correct result or concept for the given vectors.

Options: A. 13 B. √(6² + 7²) C. 1 D. 42

Correct Answer: √(6² + 7²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 6: NEET Exam-style Question - Parallelogram law formula with A = 7, B = 8, θ = 120°

Question: Parallelogram law formula. Find the correct result or concept for the given vectors.

Options: A. 15 B. √(7² + 8²) C. 1 D. 56

Correct Answer: √(57)

Detailed Explanation: For θ = 120°, cos θ = -1/2, so R² = A² + B² - AB.

NEET 7: NEET Exam-style Question - Polygon law closing condition with A = 8, B = 9, θ = 0°

Question: Polygon law closing condition. Find the correct result or concept for the given vectors.

Options: A. 17 B. √(8² + 9²) C. 1 D. 72

Correct Answer: 17

Detailed Explanation: When θ = 0°, vectors act in the same direction, so R = A + B.

NEET 8: NEET Exam-style Question - Resultant direction with A = 9, B = 3, θ = 30°

Question: Resultant direction. Find the correct result or concept for the given vectors.

Options: A. 12 B. √(9² + 3²) C. 6 D. 27

Correct Answer: √(9² + 3²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 9: NEET Exam-style Question - Horizontal and vertical velocity components with A = 10, B = 4, θ = 45°

Question: Horizontal and vertical velocity components. Find the correct result or concept for the given vectors.

Options: A. 14 B. √(10² + 4²) C. 6 D. 40

Correct Answer: √(10² + 4²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 10: NEET Exam-style Question - Force resolution on inclined plane with A = 11, B = 5, θ = 60°

Question: Force resolution on inclined plane. Find the correct result or concept for the given vectors.

Options: A. 16 B. √(11² + 5²) C. 6 D. 55

Correct Answer: √(11² + 5²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 11: NEET Exam-style Question - Resultant of two perpendicular vectors with A = 12, B = 6, θ = 90°

Question: Resultant of two perpendicular vectors. Find the correct result or concept for the given vectors.

Options: A. 18 B. √(12² + 6²) C. 6 D. 72

Correct Answer: √(12² + 6²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 12: NEET Exam-style Question - Angle between equal vectors with A = 13, B = 7, θ = 120°

Question: Angle between equal vectors. Find the correct result or concept for the given vectors.

Options: A. 20 B. √(13² + 7²) C. 6 D. 91

Correct Answer: √(127)

Detailed Explanation: For θ = 120°, cos θ = -1/2, so R² = A² + B² - AB.

NEET 13: NEET Exam-style Question - Components of an oblique vector with A = 14, B = 8, θ = 0°

Question: Components of an oblique vector. Find the correct result or concept for the given vectors.

Options: A. 22 B. √(14² + 8²) C. 6 D. 112

Correct Answer: 22

Detailed Explanation: When θ = 0°, vectors act in the same direction, so R = A + B.

NEET 14: NEET Exam-style Question - Equilibrium of three forces with A = 15, B = 9, θ = 30°

Question: Equilibrium of three forces. Find the correct result or concept for the given vectors.

Options: A. 24 B. √(15² + 9²) C. 6 D. 135

Correct Answer: √(15² + 9²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 15: NEET Exam-style Question - Triangle law in displacement with A = 16, B = 3, θ = 45°

Question: Triangle law in displacement. Find the correct result or concept for the given vectors.

Options: A. 19 B. √(16² + 3²) C. 13 D. 48

Correct Answer: √(16² + 3²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 16: NEET Exam-style Question - Parallelogram law formula with A = 17, B = 4, θ = 60°

Question: Parallelogram law formula. Find the correct result or concept for the given vectors.

Options: A. 21 B. √(17² + 4²) C. 13 D. 68

Correct Answer: √(17² + 4²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 17: NEET Exam-style Question - Polygon law closing condition with A = 18, B = 5, θ = 90°

Question: Polygon law closing condition. Find the correct result or concept for the given vectors.

Options: A. 23 B. √(18² + 5²) C. 13 D. 90

Correct Answer: √(18² + 5²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 18: NEET Exam-style Question - Resultant direction with A = 19, B = 6, θ = 120°

Question: Resultant direction. Find the correct result or concept for the given vectors.

Options: A. 25 B. √(19² + 6²) C. 13 D. 114

Correct Answer: √(283)

Detailed Explanation: For θ = 120°, cos θ = -1/2, so R² = A² + B² - AB.

NEET 19: NEET Exam-style Question - Horizontal and vertical velocity components with A = 20, B = 7, θ = 0°

Question: Horizontal and vertical velocity components. Find the correct result or concept for the given vectors.

Options: A. 27 B. √(20² + 7²) C. 13 D. 140

Correct Answer: 27

Detailed Explanation: When θ = 0°, vectors act in the same direction, so R = A + B.

NEET 20: NEET Exam-style Question - Force resolution on inclined plane with A = 21, B = 8, θ = 30°

Question: Force resolution on inclined plane. Find the correct result or concept for the given vectors.

Options: A. 29 B. √(21² + 8²) C. 13 D. 168

Correct Answer: √(21² + 8²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 21: NEET Exam-style Question - Resultant of two perpendicular vectors with A = 22, B = 9, θ = 90°

Question: Resultant of two perpendicular vectors. Find the correct result or concept for the given vectors.

Options: A. 31 B. √(22² + 9²) C. 13 D. 198

Correct Answer: √(22² + 9²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 22: NEET Exam-style Question - Angle between equal vectors with A = 23, B = 3, θ = 60°

Question: Angle between equal vectors. Find the correct result or concept for the given vectors.

Options: A. 26 B. √(23² + 3²) C. 20 D. 69

Correct Answer: √(23² + 3²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 23: NEET Exam-style Question - Components of an oblique vector with A = 24, B = 4, θ = 90°

Question: Components of an oblique vector. Find the correct result or concept for the given vectors.

Options: A. 28 B. √(24² + 4²) C. 20 D. 96

Correct Answer: √(24² + 4²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 24: NEET Exam-style Question - Equilibrium of three forces with A = 25, B = 5, θ = 120°

Question: Equilibrium of three forces. Find the correct result or concept for the given vectors.

Options: A. 30 B. √(25² + 5²) C. 20 D. 125

Correct Answer: √(525)

Detailed Explanation: For θ = 120°, cos θ = -1/2, so R² = A² + B² - AB.

NEET 25: NEET Exam-style Question - Triangle law in displacement with A = 26, B = 6, θ = 0°

Question: Triangle law in displacement. Find the correct result or concept for the given vectors.

Options: A. 32 B. √(26² + 6²) C. 20 D. 156

Correct Answer: 32

Detailed Explanation: When θ = 0°, vectors act in the same direction, so R = A + B.

NEET 26: NEET Exam-style Question - Parallelogram law formula with A = 27, B = 7, θ = 30°

Question: Parallelogram law formula. Find the correct result or concept for the given vectors.

Options: A. 34 B. √(27² + 7²) C. 20 D. 189

Correct Answer: √(27² + 7²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 27: NEET Exam-style Question - Polygon law closing condition with A = 28, B = 8, θ = 45°

Question: Polygon law closing condition. Find the correct result or concept for the given vectors.

Options: A. 36 B. √(28² + 8²) C. 20 D. 224

Correct Answer: √(28² + 8²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 28: NEET Exam-style Question - Resultant direction with A = 29, B = 9, θ = 60°

Question: Resultant direction. Find the correct result or concept for the given vectors.

Options: A. 38 B. √(29² + 9²) C. 20 D. 261

Correct Answer: √(29² + 9²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 29: NEET Exam-style Question - Horizontal and vertical velocity components with A = 30, B = 3, θ = 90°

Question: Horizontal and vertical velocity components. Find the correct result or concept for the given vectors.

Options: A. 33 B. √(30² + 3²) C. 27 D. 90

Correct Answer: √(30² + 3²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 30: NEET Exam-style Question - Force resolution on inclined plane with A = 31, B = 4, θ = 120°

Question: Force resolution on inclined plane. Find the correct result or concept for the given vectors.

Options: A. 35 B. √(31² + 4²) C. 27 D. 124

Correct Answer: √(853)

Detailed Explanation: For θ = 120°, cos θ = -1/2, so R² = A² + B² - AB.

NEET 31: NEET Exam-style Question - Resultant of two perpendicular vectors with A = 32, B = 5, θ = 90°

Question: Resultant of two perpendicular vectors. Find the correct result or concept for the given vectors.

Options: A. 37 B. √(32² + 5²) C. 27 D. 160

Correct Answer: √(32² + 5²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 32: NEET Exam-style Question - Angle between equal vectors with A = 33, B = 6, θ = 30°

Question: Angle between equal vectors. Find the correct result or concept for the given vectors.

Options: A. 39 B. √(33² + 6²) C. 27 D. 198

Correct Answer: √(33² + 6²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 33: NEET Exam-style Question - Components of an oblique vector with A = 34, B = 7, θ = 45°

Question: Components of an oblique vector. Find the correct result or concept for the given vectors.

Options: A. 41 B. √(34² + 7²) C. 27 D. 238

Correct Answer: √(34² + 7²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 34: NEET Exam-style Question - Equilibrium of three forces with A = 35, B = 8, θ = 60°

Question: Equilibrium of three forces. Find the correct result or concept for the given vectors.

Options: A. 43 B. √(35² + 8²) C. 27 D. 280

Correct Answer: √(35² + 8²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 35: NEET Exam-style Question - Triangle law in displacement with A = 36, B = 9, θ = 90°

Question: Triangle law in displacement. Find the correct result or concept for the given vectors.

Options: A. 45 B. √(36² + 9²) C. 27 D. 324

Correct Answer: √(36² + 9²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 36: NEET Exam-style Question - Parallelogram law formula with A = 37, B = 3, θ = 120°

Question: Parallelogram law formula. Find the correct result or concept for the given vectors.

Options: A. 40 B. √(37² + 3²) C. 34 D. 111

Correct Answer: √(1267)

Detailed Explanation: For θ = 120°, cos θ = -1/2, so R² = A² + B² - AB.

NEET 37: NEET Exam-style Question - Polygon law closing condition with A = 38, B = 4, θ = 0°

Question: Polygon law closing condition. Find the correct result or concept for the given vectors.

Options: A. 42 B. √(38² + 4²) C. 34 D. 152

Correct Answer: 42

Detailed Explanation: When θ = 0°, vectors act in the same direction, so R = A + B.

NEET 38: NEET Exam-style Question - Resultant direction with A = 39, B = 5, θ = 30°

Question: Resultant direction. Find the correct result or concept for the given vectors.

Options: A. 44 B. √(39² + 5²) C. 34 D. 195

Correct Answer: √(39² + 5²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 39: NEET Exam-style Question - Horizontal and vertical velocity components with A = 40, B = 6, θ = 45°

Question: Horizontal and vertical velocity components. Find the correct result or concept for the given vectors.

Options: A. 46 B. √(40² + 6²) C. 34 D. 240

Correct Answer: √(40² + 6²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 40: NEET Exam-style Question - Force resolution on inclined plane with A = 41, B = 7, θ = 60°

Question: Force resolution on inclined plane. Find the correct result or concept for the given vectors.

Options: A. 48 B. √(41² + 7²) C. 34 D. 287

Correct Answer: √(41² + 7²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 41: NEET Exam-style Question - Resultant of two perpendicular vectors with A = 42, B = 8, θ = 90°

Question: Resultant of two perpendicular vectors. Find the correct result or concept for the given vectors.

Options: A. 50 B. √(42² + 8²) C. 34 D. 336

Correct Answer: √(42² + 8²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 42: NEET Exam-style Question - Angle between equal vectors with A = 43, B = 9, θ = 120°

Question: Angle between equal vectors. Find the correct result or concept for the given vectors.

Options: A. 52 B. √(43² + 9²) C. 34 D. 387

Correct Answer: √(1543)

Detailed Explanation: For θ = 120°, cos θ = -1/2, so R² = A² + B² - AB.

NEET 43: NEET Exam-style Question - Components of an oblique vector with A = 44, B = 3, θ = 0°

Question: Components of an oblique vector. Find the correct result or concept for the given vectors.

Options: A. 47 B. √(44² + 3²) C. 41 D. 132

Correct Answer: 47

Detailed Explanation: When θ = 0°, vectors act in the same direction, so R = A + B.

NEET 44: NEET Exam-style Question - Equilibrium of three forces with A = 45, B = 4, θ = 30°

Question: Equilibrium of three forces. Find the correct result or concept for the given vectors.

Options: A. 49 B. √(45² + 4²) C. 41 D. 180

Correct Answer: √(45² + 4²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 45: NEET Exam-style Question - Triangle law in displacement with A = 46, B = 5, θ = 45°

Question: Triangle law in displacement. Find the correct result or concept for the given vectors.

Options: A. 51 B. √(46² + 5²) C. 41 D. 230

Correct Answer: √(46² + 5²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 46: NEET Exam-style Question - Parallelogram law formula with A = 47, B = 6, θ = 60°

Question: Parallelogram law formula. Find the correct result or concept for the given vectors.

Options: A. 53 B. √(47² + 6²) C. 41 D. 282

Correct Answer: √(47² + 6²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 47: NEET Exam-style Question - Polygon law closing condition with A = 48, B = 7, θ = 90°

Question: Polygon law closing condition. Find the correct result or concept for the given vectors.

Options: A. 55 B. √(48² + 7²) C. 41 D. 336

Correct Answer: √(48² + 7²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

NEET 48: NEET Exam-style Question - Resultant direction with A = 49, B = 8, θ = 120°

Question: Resultant direction. Find the correct result or concept for the given vectors.

Options: A. 57 B. √(49² + 8²) C. 41 D. 392

Correct Answer: √(2073)

Detailed Explanation: For θ = 120°, cos θ = -1/2, so R² = A² + B² - AB.

NEET 49: NEET Exam-style Question - Horizontal and vertical velocity components with A = 50, B = 9, θ = 0°

Question: Horizontal and vertical velocity components. Find the correct result or concept for the given vectors.

Options: A. 59 B. √(50² + 9²) C. 41 D. 450

Correct Answer: 59

Detailed Explanation: When θ = 0°, vectors act in the same direction, so R = A + B.

NEET 50: NEET Exam-style Question - Force resolution on inclined plane with A = 51, B = 3, θ = 30°

Question: Force resolution on inclined plane. Find the correct result or concept for the given vectors.

Options: A. 54 B. √(51² + 3²) C. 48 D. 153

Correct Answer: √(51² + 3²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main PYQ Section

50 high-quality JEE Main exam-style questions on components, resultant and equilibrium.

JEE Main 1: JEE Main Exam-style Question - Vector addition by components with A = 2, B = 3, θ = 0°

Question: Vector addition by components. Find the correct result or concept for the given vectors.

Options: A. 5 B. √(2² + 3²) C. 1 D. 6

Correct Answer: 5

Detailed Explanation: When θ = 0°, vectors act in the same direction, so R = A + B.

JEE Main 2: JEE Main Exam-style Question - Resultant from two inclined vectors with A = 3, B = 4, θ = 30°

Question: Resultant from two inclined vectors. Find the correct result or concept for the given vectors.

Options: A. 7 B. √(3² + 4²) C. 1 D. 12

Correct Answer: √(3² + 4²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 3: JEE Main Exam-style Question - Equilibrant of a force system with A = 4, B = 5, θ = 45°

Question: Equilibrant of a force system. Find the correct result or concept for the given vectors.

Options: A. 9 B. √(4² + 5²) C. 1 D. 20

Correct Answer: √(4² + 5²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 4: JEE Main Exam-style Question - Find unknown component with A = 5, B = 6, θ = 60°

Question: Find unknown component. Find the correct result or concept for the given vectors.

Options: A. 11 B. √(5² + 6²) C. 1 D. 30

Correct Answer: √(5² + 6²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 5: JEE Main Exam-style Question - Analytical method for three vectors with A = 6, B = 7, θ = 90°

Question: Analytical method for three vectors. Find the correct result or concept for the given vectors.

Options: A. 13 B. √(6² + 7²) C. 1 D. 42

Correct Answer: √(6² + 7²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 6: JEE Main Exam-style Question - Direction of resultant with A = 7, B = 8, θ = 120°

Question: Direction of resultant. Find the correct result or concept for the given vectors.

Options: A. 15 B. √(7² + 8²) C. 1 D. 56

Correct Answer: √(57)

Detailed Explanation: For θ = 120°, cos θ = -1/2, so R² = A² + B² - AB.

JEE Main 7: JEE Main Exam-style Question - Closed polygon condition with A = 8, B = 9, θ = 0°

Question: Closed polygon condition. Find the correct result or concept for the given vectors.

Options: A. 17 B. √(8² + 9²) C. 1 D. 72

Correct Answer: 17

Detailed Explanation: When θ = 0°, vectors act in the same direction, so R = A + B.

JEE Main 8: JEE Main Exam-style Question - Component sum zero with A = 9, B = 3, θ = 30°

Question: Component sum zero. Find the correct result or concept for the given vectors.

Options: A. 12 B. √(9² + 3²) C. 6 D. 27

Correct Answer: √(9² + 3²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 9: JEE Main Exam-style Question - Relative displacement by vectors with A = 10, B = 4, θ = 45°

Question: Relative displacement by vectors. Find the correct result or concept for the given vectors.

Options: A. 14 B. √(10² + 4²) C. 6 D. 40

Correct Answer: √(10² + 4²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 10: JEE Main Exam-style Question - Special angle vector addition with A = 11, B = 5, θ = 60°

Question: Special angle vector addition. Find the correct result or concept for the given vectors.

Options: A. 16 B. √(11² + 5²) C. 6 D. 55

Correct Answer: √(11² + 5²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 11: JEE Main Exam-style Question - Vector addition by components with A = 12, B = 6, θ = 90°

Question: Vector addition by components. Find the correct result or concept for the given vectors.

Options: A. 18 B. √(12² + 6²) C. 6 D. 72

Correct Answer: √(12² + 6²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 12: JEE Main Exam-style Question - Resultant from two inclined vectors with A = 13, B = 7, θ = 120°

Question: Resultant from two inclined vectors. Find the correct result or concept for the given vectors.

Options: A. 20 B. √(13² + 7²) C. 6 D. 91

Correct Answer: √(127)

Detailed Explanation: For θ = 120°, cos θ = -1/2, so R² = A² + B² - AB.

JEE Main 13: JEE Main Exam-style Question - Equilibrant of a force system with A = 14, B = 8, θ = 0°

Question: Equilibrant of a force system. Find the correct result or concept for the given vectors.

Options: A. 22 B. √(14² + 8²) C. 6 D. 112

Correct Answer: 22

Detailed Explanation: When θ = 0°, vectors act in the same direction, so R = A + B.

JEE Main 14: JEE Main Exam-style Question - Find unknown component with A = 15, B = 9, θ = 30°

Question: Find unknown component. Find the correct result or concept for the given vectors.

Options: A. 24 B. √(15² + 9²) C. 6 D. 135

Correct Answer: √(15² + 9²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 15: JEE Main Exam-style Question - Analytical method for three vectors with A = 16, B = 3, θ = 45°

Question: Analytical method for three vectors. Find the correct result or concept for the given vectors.

Options: A. 19 B. √(16² + 3²) C. 13 D. 48

Correct Answer: √(16² + 3²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 16: JEE Main Exam-style Question - Direction of resultant with A = 17, B = 4, θ = 60°

Question: Direction of resultant. Find the correct result or concept for the given vectors.

Options: A. 21 B. √(17² + 4²) C. 13 D. 68

Correct Answer: √(17² + 4²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 17: JEE Main Exam-style Question - Closed polygon condition with A = 18, B = 5, θ = 90°

Question: Closed polygon condition. Find the correct result or concept for the given vectors.

Options: A. 23 B. √(18² + 5²) C. 13 D. 90

Correct Answer: √(18² + 5²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 18: JEE Main Exam-style Question - Component sum zero with A = 19, B = 6, θ = 120°

Question: Component sum zero. Find the correct result or concept for the given vectors.

Options: A. 25 B. √(19² + 6²) C. 13 D. 114

Correct Answer: √(283)

Detailed Explanation: For θ = 120°, cos θ = -1/2, so R² = A² + B² - AB.

JEE Main 19: JEE Main Exam-style Question - Relative displacement by vectors with A = 20, B = 7, θ = 0°

Question: Relative displacement by vectors. Find the correct result or concept for the given vectors.

Options: A. 27 B. √(20² + 7²) C. 13 D. 140

Correct Answer: 27

Detailed Explanation: When θ = 0°, vectors act in the same direction, so R = A + B.

JEE Main 20: JEE Main Exam-style Question - Special angle vector addition with A = 21, B = 8, θ = 30°

Question: Special angle vector addition. Find the correct result or concept for the given vectors.

Options: A. 29 B. √(21² + 8²) C. 13 D. 168

Correct Answer: √(21² + 8²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 21: JEE Main Exam-style Question - Vector addition by components with A = 22, B = 9, θ = 45°

Question: Vector addition by components. Find the correct result or concept for the given vectors.

Options: A. 31 B. √(22² + 9²) C. 13 D. 198

Correct Answer: √(22² + 9²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 22: JEE Main Exam-style Question - Resultant from two inclined vectors with A = 23, B = 3, θ = 60°

Question: Resultant from two inclined vectors. Find the correct result or concept for the given vectors.

Options: A. 26 B. √(23² + 3²) C. 20 D. 69

Correct Answer: √(23² + 3²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 23: JEE Main Exam-style Question - Equilibrant of a force system with A = 24, B = 4, θ = 90°

Question: Equilibrant of a force system. Find the correct result or concept for the given vectors.

Options: A. 28 B. √(24² + 4²) C. 20 D. 96

Correct Answer: √(24² + 4²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 24: JEE Main Exam-style Question - Find unknown component with A = 25, B = 5, θ = 120°

Question: Find unknown component. Find the correct result or concept for the given vectors.

Options: A. 30 B. √(25² + 5²) C. 20 D. 125

Correct Answer: √(525)

Detailed Explanation: For θ = 120°, cos θ = -1/2, so R² = A² + B² - AB.

JEE Main 25: JEE Main Exam-style Question - Analytical method for three vectors with A = 26, B = 6, θ = 0°

Question: Analytical method for three vectors. Find the correct result or concept for the given vectors.

Options: A. 32 B. √(26² + 6²) C. 20 D. 156

Correct Answer: 32

Detailed Explanation: When θ = 0°, vectors act in the same direction, so R = A + B.

JEE Main 26: JEE Main Exam-style Question - Direction of resultant with A = 27, B = 7, θ = 30°

Question: Direction of resultant. Find the correct result or concept for the given vectors.

Options: A. 34 B. √(27² + 7²) C. 20 D. 189

Correct Answer: √(27² + 7²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 27: JEE Main Exam-style Question - Closed polygon condition with A = 28, B = 8, θ = 45°

Question: Closed polygon condition. Find the correct result or concept for the given vectors.

Options: A. 36 B. √(28² + 8²) C. 20 D. 224

Correct Answer: √(28² + 8²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 28: JEE Main Exam-style Question - Component sum zero with A = 29, B = 9, θ = 60°

Question: Component sum zero. Find the correct result or concept for the given vectors.

Options: A. 38 B. √(29² + 9²) C. 20 D. 261

Correct Answer: √(29² + 9²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 29: JEE Main Exam-style Question - Relative displacement by vectors with A = 30, B = 3, θ = 90°

Question: Relative displacement by vectors. Find the correct result or concept for the given vectors.

Options: A. 33 B. √(30² + 3²) C. 27 D. 90

Correct Answer: √(30² + 3²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 30: JEE Main Exam-style Question - Special angle vector addition with A = 31, B = 4, θ = 120°

Question: Special angle vector addition. Find the correct result or concept for the given vectors.

Options: A. 35 B. √(31² + 4²) C. 27 D. 124

Correct Answer: √(853)

Detailed Explanation: For θ = 120°, cos θ = -1/2, so R² = A² + B² - AB.

JEE Main 31: JEE Main Exam-style Question - Vector addition by components with A = 32, B = 5, θ = 0°

Question: Vector addition by components. Find the correct result or concept for the given vectors.

Options: A. 37 B. √(32² + 5²) C. 27 D. 160

Correct Answer: 37

Detailed Explanation: When θ = 0°, vectors act in the same direction, so R = A + B.

JEE Main 32: JEE Main Exam-style Question - Resultant from two inclined vectors with A = 33, B = 6, θ = 30°

Question: Resultant from two inclined vectors. Find the correct result or concept for the given vectors.

Options: A. 39 B. √(33² + 6²) C. 27 D. 198

Correct Answer: √(33² + 6²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 33: JEE Main Exam-style Question - Equilibrant of a force system with A = 34, B = 7, θ = 45°

Question: Equilibrant of a force system. Find the correct result or concept for the given vectors.

Options: A. 41 B. √(34² + 7²) C. 27 D. 238

Correct Answer: √(34² + 7²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 34: JEE Main Exam-style Question - Find unknown component with A = 35, B = 8, θ = 60°

Question: Find unknown component. Find the correct result or concept for the given vectors.

Options: A. 43 B. √(35² + 8²) C. 27 D. 280

Correct Answer: √(35² + 8²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 35: JEE Main Exam-style Question - Analytical method for three vectors with A = 36, B = 9, θ = 90°

Question: Analytical method for three vectors. Find the correct result or concept for the given vectors.

Options: A. 45 B. √(36² + 9²) C. 27 D. 324

Correct Answer: √(36² + 9²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 36: JEE Main Exam-style Question - Direction of resultant with A = 37, B = 3, θ = 120°

Question: Direction of resultant. Find the correct result or concept for the given vectors.

Options: A. 40 B. √(37² + 3²) C. 34 D. 111

Correct Answer: √(1267)

Detailed Explanation: For θ = 120°, cos θ = -1/2, so R² = A² + B² - AB.

JEE Main 37: JEE Main Exam-style Question - Closed polygon condition with A = 38, B = 4, θ = 0°

Question: Closed polygon condition. Find the correct result or concept for the given vectors.

Options: A. 42 B. √(38² + 4²) C. 34 D. 152

Correct Answer: 42

Detailed Explanation: When θ = 0°, vectors act in the same direction, so R = A + B.

JEE Main 38: JEE Main Exam-style Question - Component sum zero with A = 39, B = 5, θ = 30°

Question: Component sum zero. Find the correct result or concept for the given vectors.

Options: A. 44 B. √(39² + 5²) C. 34 D. 195

Correct Answer: √(39² + 5²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 39: JEE Main Exam-style Question - Relative displacement by vectors with A = 40, B = 6, θ = 45°

Question: Relative displacement by vectors. Find the correct result or concept for the given vectors.

Options: A. 46 B. √(40² + 6²) C. 34 D. 240

Correct Answer: √(40² + 6²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 40: JEE Main Exam-style Question - Special angle vector addition with A = 41, B = 7, θ = 60°

Question: Special angle vector addition. Find the correct result or concept for the given vectors.

Options: A. 48 B. √(41² + 7²) C. 34 D. 287

Correct Answer: √(41² + 7²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 41: JEE Main Exam-style Question - Vector addition by components with A = 42, B = 8, θ = 90°

Question: Vector addition by components. Find the correct result or concept for the given vectors.

Options: A. 50 B. √(42² + 8²) C. 34 D. 336

Correct Answer: √(42² + 8²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 42: JEE Main Exam-style Question - Resultant from two inclined vectors with A = 43, B = 9, θ = 120°

Question: Resultant from two inclined vectors. Find the correct result or concept for the given vectors.

Options: A. 52 B. √(43² + 9²) C. 34 D. 387

Correct Answer: √(1543)

Detailed Explanation: For θ = 120°, cos θ = -1/2, so R² = A² + B² - AB.

JEE Main 43: JEE Main Exam-style Question - Equilibrant of a force system with A = 44, B = 3, θ = 0°

Question: Equilibrant of a force system. Find the correct result or concept for the given vectors.

Options: A. 47 B. √(44² + 3²) C. 41 D. 132

Correct Answer: 47

Detailed Explanation: When θ = 0°, vectors act in the same direction, so R = A + B.

JEE Main 44: JEE Main Exam-style Question - Find unknown component with A = 45, B = 4, θ = 30°

Question: Find unknown component. Find the correct result or concept for the given vectors.

Options: A. 49 B. √(45² + 4²) C. 41 D. 180

Correct Answer: √(45² + 4²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 45: JEE Main Exam-style Question - Analytical method for three vectors with A = 46, B = 5, θ = 45°

Question: Analytical method for three vectors. Find the correct result or concept for the given vectors.

Options: A. 51 B. √(46² + 5²) C. 41 D. 230

Correct Answer: √(46² + 5²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 46: JEE Main Exam-style Question - Direction of resultant with A = 47, B = 6, θ = 60°

Question: Direction of resultant. Find the correct result or concept for the given vectors.

Options: A. 53 B. √(47² + 6²) C. 41 D. 282

Correct Answer: √(47² + 6²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 47: JEE Main Exam-style Question - Closed polygon condition with A = 48, B = 7, θ = 90°

Question: Closed polygon condition. Find the correct result or concept for the given vectors.

Options: A. 55 B. √(48² + 7²) C. 41 D. 336

Correct Answer: √(48² + 7²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Main 48: JEE Main Exam-style Question - Component sum zero with A = 49, B = 8, θ = 120°

Question: Component sum zero. Find the correct result or concept for the given vectors.

Options: A. 57 B. √(49² + 8²) C. 41 D. 392

Correct Answer: √(2073)

Detailed Explanation: For θ = 120°, cos θ = -1/2, so R² = A² + B² - AB.

JEE Main 49: JEE Main Exam-style Question - Relative displacement by vectors with A = 50, B = 9, θ = 0°

Question: Relative displacement by vectors. Find the correct result or concept for the given vectors.

Options: A. 59 B. √(50² + 9²) C. 41 D. 450

Correct Answer: 59

Detailed Explanation: When θ = 0°, vectors act in the same direction, so R = A + B.

JEE Main 50: JEE Main Exam-style Question - Special angle vector addition with A = 51, B = 3, θ = 30°

Question: Special angle vector addition. Find the correct result or concept for the given vectors.

Options: A. 54 B. √(51² + 3²) C. 48 D. 153

Correct Answer: √(51² + 3²)

Detailed Explanation: Use component method or resultant formula. For perpendicular vectors, R = √(A² + B²). For inclined vectors, use R = √(A² + B² + 2AB cos θ).

JEE Advanced PYQ Section

50 difficult conceptual and numerical JEE Advanced exam-style questions with solutions.

JEE Advanced 1: Multi-vector geometry with constraints

Question: Multi-vector geometry with constraints

Answer: Resolve all given vectors into x and y components, add components algebraically, then use R = √(R_x² + R_y²) and tan θ = R_y / R_x.