Vector Addition – Triangle and Parallelogram Law - UNSOLVED PRACTICE SET
Chapter: Kinematics | Topic: Vector Addition Triangle and Parallelogram Law
VECTOR ADDITION – TRIANGLE AND PARALLELOGRAM LAW - UNSOLVED PRACTICE SET
Topic: Vector Addition Triangle and Parallelogram Law
Multiple Choice Questions
Q1. According to the triangle law of vector addition, if two vectors are represented by two sides of a triangle taken in order, their resultant is represented by:
- The third side taken in the opposite order
- The third side taken in the same order
- Half of the third side
- Twice the third side
Q2. The magnitude of the resultant of two vectors A and B with angle θ between them is:
- √(A² + B² + 2AB cos θ)
- √(A² + B² − 2AB cos θ)
- A + B
- A − B
Q3. Two vectors of equal magnitude A have an angle of 120° between them. Their resultant has magnitude:
- 2A
- A
- A√3
- 0
Q4. In the parallelogram law of vector addition, the resultant of two vectors is represented by:
- A diagonal of the parallelogram
- A side of the parallelogram
- The area of the parallelogram
- The perimeter of the parallelogram
Q5. Two vectors are perpendicular to each other. The angle between their resultant and one of the vectors is given by:
- tan⁻¹(A/B)
- tan⁻¹(B/A)
- 45°
- 90°
Q6. The resultant of two vectors is maximum when the angle between them is:
- 0°
- 90°
- 180°C
- 45°
Short Answer Questions
Q7. State the triangle law of vector addition. Illustrate it with a diagram.
Q8. State the parallelogram law of vector addition. How is it related to the triangle law?
Q9. Two vectors of magnitudes 6 N and 8 N act at right angles to each other. Calculate the magnitude and direction of their resultant.
Q10. In a school tug-of-war, two teams pull a rope with forces of 500 N and 400 N at an angle of 60° to each other. Calculate the resultant force on the rope.
Q11. Two vectors A and B have a resultant R. If the magnitude of B is doubled, the new resultant becomes perpendicular to A. Find the relation between the magnitudes of A and B.
Q12. Prove that the magnitude of the resultant of two vectors is always less than or equal to the sum of their magnitudes and greater than or equal to the difference of their magnitudes.
Long Answer Questions
Q13. Derive the analytical expression for the magnitude and direction of the resultant of two vectors using:
(i) The parallelogram law
(ii) The triangle law
Show that both methods give the same result.
Q14. Two forces F₁ and F₂ act on a particle. F₁ has magnitude 10 N and acts along the positive x-axis. F₂ has magnitude 15 N and acts at 120° to F₁.
(a) Find the resultant force using the parallelogram law.
(b) Verify your result by resolving the vectors into components.
(c) A third force F₃ is applied such that the particle is in equilibrium. Find F₃.
Q15. Three vectors A, B, and C have magnitudes 5 units, 5 units, and 5√2 units respectively. Vector A is along the x-axis, Vector B is at 90° to A, and Vector C is the resultant of A and B.
(a) Verify using the parallelogram law that C = A + B.
(b) Calculate the angle between A and C.
(c) If a fourth vector D is added such that A + B + C + D = 0, find D.
Application-Based Problems
Q16. A boat is moving at 4 m/s relative to water in a direction 30° east of north. The river flows at 2 m/s due east.
(a) Draw a vector diagram showing the velocities.
(b) Use the parallelogram law to find the resultant velocity of the boat relative to the ground.
(c) Calculate the magnitude and direction of the resultant velocity.
(d) Verify your answer using the component method.
Q17. Two tractors pull a stuck truck with forces of 8000 N and 6000 N. The angle between the ropes is 50°.
(a) Calculate the resultant force on the truck using the law of cosines.
(b) Calculate the direction of the resultant force with respect to the 8000 N force.
(c) If a third tractor pulls with force F at 120° to the 8000 N force, what should F be so that the resultant is along the direction of the 8000 N force?
Q18. In a physics experiment, three forces are applied to a ring at the centre of a force table: F₁ = 5 N at 0°, F₂ = 5 N at 120°, and F₃ = 5 N at 240°.
(a) Calculate the resultant of F₁ and F₂ using the parallelogram law.
(b) Calculate the resultant of this result with F₃.
(c) What is the net force on the ring? Explain your observation.
(d) What does this experiment demonstrate about vector addition?