Scalars and Vectors Problems - Physics 11th - Chapter 2 - Sindh Textbook Board

  

SCALARS AND VECTORS 

(PROBLEMS)

1. State which of the following are scalars and which are vectors.

2. Find the resultant of the following displacement:

A = 20km 30° south of east;

B = 50km due west

C = 40km north east;

D = 30km 60° south of west

3. An aeroplane flies 400km due west from city A to city B, then 300km north east to city C, and finally 100km north to city D, How far is it from city A to D? In what direction must the aeroplane had to return directly to city A from city D?

4. Show graphically that -(A - B) = -A + B

5. Given vector A.B.C and D as shown in figure below. Construct (a) 4A - 3B - (2C + 2D) (b) (1/2)(C) + (1/3)(A + B + 2D)

6. The following forces act on a particle P:F1→ = 2i + 3j - 5k, F2→ = -5i + j + 3k F3→ = i - 2j + 4k, F4→ = 4i - 3j - 2k measured in newtons Find (a) the resultant of the forces (b) the magnitude of the resultant force

7. If A = 3i - j - 4k, B = -2i + 4j - 3k, C = i + 2j - k, find (a) 2A - B + 3C, (b) |A + B + C|, (c) |3A - 2B + 4C| (d) a unit vector parallel to 3A - 2B + 4C  

8. Two tugboats are towing a ship, Each exerts a force of 6000N, and angle between the two ropes is 60°. Calculate the resultant force on the ship. 

9. The position vectors of point P and Q are given by r1 = 2i + 3j - k, r2 = 4i - 3j + 3k. Determine PQ in terms of rectangular unit vector i, j, and k and find its magnitude.

10. Prove that the vectors A = 3i + j - 2k, B = -i + 3j + 4k, C = 4i - 2j - 6k, can form the sides of a triangle. Find the length of the medians of the triangle.

11. Find the rectangular components of a vector A, 15unit long when it form an angle with respect to +ve  x-axis of (i) 50° (ii) 130° (iii) 230°, (iv) 310°

12. Two vectors 10cm and 8cm long form an angle of (a) 60°, (b)90° and (c) 120°. Find the magnitude of difference and the angle with respect to the larger vector.

13. The angle between the vector A and B is 60°. Given that |A| = |B| = 1, calculate (a) |B-A|; (b)|B + A|

14. A car weighing 10,000N on a hill which makes an angle of 20° with the horizontal. Find the components of car's weight parallel and perpendicular to the road.

15. Find the angle between A = 2i + 2j - k and B = 6i - 3j + 2K.

16.Find the projection of the vector. A = i - 2j + k onto the direction of vector B = 4i - 4j + 7k.

17. Find the angle, α, β, γ which the vector A = 3i - 6j + 2k makes with the positive x, y, z axis respectively.

18. Find the work done in moving an object along a vector r = 3i + 2j - 5k if the applied force is F = 2i - j - k.

19. Find the work done by a force of 30,000N in moving an object through a distance of 45m when: (a) the force is in the direction of motion, and (b) the force makes an angle of 40° to the direction of motion. Find the rate at which the force is working at a time when the velocity is 2m/sec.

20. Two vectors A and B are such that |A| = 3, |B| = 4, and A.B = -5, find:

(a) the angle between A and B

(b) the length of |A+B| and |A-B|

(c) the angle between (A + B) and (A - B).

21. If A = 2i - 3j - k, B = i + 4j - 2k. Find (a) A x B (b) B x A, and (c) (A + B) x (A - B)

22. Determine the unit vector perpendicular to the plane A = 2i - 6j - 3k and B = 4i + 3j - k

23. Using the definition of vector product, prove the law of sines for plane triangles of sides a, b, and c.

24. If r1 and r2 are the position vectors (both lie in xy plane) making angles θ1 and θ2 with the position x-axis measured counterclockwise, find their vector product when:

(i) r1 = 4cm with θ1 = 30°    r2 = 3cm with θ2 = 90°

(ii) r1 = 6cm with θ1 = 220°  r2 = 3cm with θ2 = 40°

(iii) r1 = 10cm with θ1 = 20°  r2 = 9cm with θ2 = 110°