Resolution of Vectors and Unit Vectors - UNSOLVED PRACTICE SET
Chapter: Kinematics | Topic: Resolution of Vectors and Unit Vectors
RESOLUTION OF VECTORS AND UNIT VECTORS - UNSOLVED PRACTICE SET
Topic: Resolution of Vectors and Unit Vectors
SECTION NAME
Q1. The rectangular components of a vector A making angle θ with the x-axis are:
- A cos θ and A sin θ
- A sin θ and A cos θ
- A tan θ and A cot θ
- A sec θ and A cosec θ
Q2. The unit vector along the x-axis is denoted by:
- î
- j
- k
- r̂
Q3. If A = 3î + 4ĵ, then the unit vector in the direction of A is:
- (3î + 4ĵ)/5
- 3î + 4j
- (3î + 4ĵ)/25
- 5(3î + 4ĵ)
Q4. A vector can be resolved into components along:
- Only two perpendicular directions
- Any two or more directions
- Only the x and y directions
- Only one direction
Q5. The magnitude of the vector 6î − 8ĵ is:
- 2
- 10
- 14
- 100
Q6. If A = Aₓî + Aᵧĵ, then the angle θ that A makes with the x-axis is given by:
- tan θ = Aₓ/Aᵧ
- tan θ = Aᵧ/Aₓ
- tan θ = Aₓ + Aᵧ
- tan θ = Aₓ − Aᵧ
Short Answer Questions
Q7. Define resolution of a vector. Explain why rectangular resolution is most commonly used.
Q8. A vector A has magnitude 10 units and makes an angle of 60° with the positive x-axis. Find its rectangular components.
Q9. Express the vector B = 5î − 12ĵ in polar form (magnitude and direction).
Q10. In your school, a student pulls a box with a force of 50 N at an angle of 30° above the horizontal. Resolve this force into horizontal and vertical components.
Q11. Two vectors A = 2î + 3ĵ and B = 4î − ĵ are given. Find:
(a) A + B
(b) A − B
(c) 2A + 3B
Q12. What are unit vectors? Write the unit vectors along the x, y, and z axes. How is a unit vector in the direction of any given vector obtained?
Long Answer Questions
Q13. Explain the method of resolution of a vector into rectangular components. Derive expressions for the components of a vector A in two dimensions. Show how the original vector can be reconstructed from its components.
Q14. A force F = 100 N acts on a body at an angle of 37° with the horizontal.
(a) Resolve F into horizontal and vertical components.
(b) Calculate the work done by the horizontal component if the body moves 10 m horizontally.
(c) Calculate the work done by the vertical component if the body moves 10 m horizontally. (Hint: Work = F·s = Fs cos θ)
(d) What is the total work done by the force F?
Q15. Three unit vectors â, b̂, and ĉ are directed along the sides of an equilateral triangle. Vector â is along the x-axis.
(a) Express b̂ and ĉ in terms of î and ĵ.
(b) Verify that â + b̂ + ĉ = 0.
(c) What does this result tell you about the vector sum of three equal vectors at 120° to each other?
Application-Based Problems
Q16. A projectile is launched with a velocity of 50 m/s at an angle of 60° with the horizontal.
(a) Resolve the initial velocity into horizontal and vertical components.
(b) Calculate the horizontal distance travelled in 2 seconds.
(c) Calculate the vertical height reached after 2 seconds.
(d) At what time will the vertical component of velocity become zero?
(e) What will be the horizontal component of velocity at that instant?
Q17. In a physics lab, a student applies three forces to a point object: F₁ = 10 N at 30° to the x-axis, F₂ = 15 N at 120° to the x-axis, and F₃ = 20 N at 240° to the x-axis.
(a) Resolve each force into x and y components.
(b) Calculate the net force in the x-direction.
(c) Calculate the net force in the y-direction.
(d) Find the magnitude and direction of the resultant force.
Q18. A vector A in three dimensions makes angles α, β, and γ with the x, y, and z axes respectively. These are called direction cosines.
(a) If A = 3î + 4ĵ + 12k̂, calculate the direction cosines.
(b) Verify that cos²α + cos²β + cos²γ = 1.
(c) Calculate the angle that A makes with each coordinate axis.