VECTORS REVISION QUESTIONS WITH ANSWERS

1. Express vector \( \vec{a} = (3, 4) \) in terms of \( \vec{i} \) and \( \vec{j} \).

2. Given \( \vec{a} = 4\vec{i} + 3\vec{j} \) and \( \vec{b} = -2\vec{i} + \vec{j} \), find \( \vec{a} + \vec{b} \).

3. Find the magnitude of vector \( \vec{v} = 6\vec{i} - 8\vec{j} \).

4. If \( \vec{a} = 2\vec{i} + 5\vec{j} \), find \( 3\vec{a} \).

5. A displacement vector moves 5 units East and 12 units North. Represent it using \( \vec{i} \) and \( \vec{j} \).

6. Given vectors \( \vec{u} = 3\vec{i} - 2\vec{j} \), \( \vec{v} = -\vec{i} + 4\vec{j} \), compute \( \vec{u} - \vec{v} \).

7. Vector \( \vec{r} = a\vec{i} + b\vec{j} \) has magnitude 10. If \( a = 6 \), find \( b \).

8. If \( \vec{a} = \vec{i} + 2\vec{j} \), and \( \vec{b} = 2\vec{i} - 3\vec{j} \), find \( 2\vec{a} + \vec{b} \).

9. A triangle has vectors \( \vec{AB} = 2\vec{i} + \vec{j} \), and \( \vec{BC} = -\vec{i} + 4\vec{j} \), find \( \vec{AC} \).

10. Represent a vector of magnitude 10 at 45° using \( \vec{i} \) and \( \vec{j} \).

11. Find a unit vector in the direction of \( \vec{v} = 3\vec{i} + 4\vec{j} \).

12. Find the dot product of \( \vec{a} = 5\vec{i} + 2\vec{j} \) and \( \vec{b} = \vec{i} - 3\vec{j} \).

13. Two vectors form adjacent sides of a triangle: \( \vec{AB} = \vec{i} + 2\vec{j} \) and \( \vec{AC} = 2\vec{i} - \vec{j} \). Find \( \vec{BC} \).

14. If \( \vec{p} = 2\vec{i} - 2\vec{j} \), find its direction (angle with x-axis).

15. A velocity vector is \( \vec{v} = 60\vec{i} + 20\vec{j} \) m/s. Find the speed.

16. If \( \vec{F}_1 = 3\vec{i} + \vec{j} \) and \( \vec{F}_2 = -\vec{i} + 4\vec{j} \), find the resultant force.

17. Find a vector perpendicular to \( \vec{v} = 2\vec{i} + 3\vec{j} \).

18. Resolve the vector \( 5\vec{i} + 5\vec{j} \) into a magnitude and direction.

19. Find the midpoint of the vector from \( A(1,2) \) to \( B(5,6) \).

20. If \( \vec{v} = \vec{i} + 2\vec{j} \), find a scalar \( k \) such that \( k\vec{v} = -3\vec{i} -6\vec{j} \).

21. Find the displacement vector from \( P(0,0) \) to \( Q(7,-3) \).

22. Show that \( \vec{a} = \vec{i} + \vec{j} \) and \( \vec{b} = -\vec{j} + \vec{i} \) are perpendicular.

23. Express the position vector of point \( C \) relative to origin if \( C \) lies 6 units right and 8 units down.

24. Find the angle between vectors \( \vec{a} = 2\vec{i} \) and \( \vec{b} = 3\vec{i} + 4\vec{j} \).

25. If a car moves along \( \vec{d} = 10\vec{i} + 6\vec{j} \) km, find total distance moved.

26. Two forces \( \vec{F}_1 = 4\vec{i} \), \( \vec{F}_2 = 3\vec{j} \) act together. Find the resultant vector and its magnitude.

27. Find a unit vector in the direction of \( \vec{v} = -7\vec{i} + 24\vec{j} \).

28. Determine if \( \vec{a} = 2\vec{i} + 3\vec{j} \) and \( \vec{b} = 4\vec{i} + 6\vec{j} \) are parallel.

29. If \( \vec{r} = t(2\vec{i} - \vec{j}) \), find \( \vec{r} \) when \( t = -3 \).

30. A triangle has vertices \( A(0,0), B(4,3), C(6,1) \). Find vectors \( \vec{AB} \), \( \vec{BC} \), and \( \vec{AC} \).

Suggested Answers:

1. \( 3\vec{i} + 4\vec{j} \)
2. \( (4 - 2)\vec{i} + (3 + 1)\vec{j} = 2\vec{i} + 4\vec{j} \)
3. \( \sqrt{6^2 + (-8)^2} = 10 \)
4. \( 6\vec{i} + 15\vec{j} \)
5. \( 5\vec{i} + 12\vec{j} \)
6. \( 4\vec{i} - 6\vec{j} \)
7. \( b = 8 \) since \( \sqrt{6^2 + b^2} = 10 \)
8. \( 2\vec{i} + \vec{j} \)
9. \( \vec{AC} = \vec{AB} + \vec{BC} = \vec{i} + 5\vec{j} \)
10. \( 10(\cos45^\circ \vec{i} + \sin45^\circ \vec{j}) = 7.07\vec{i} + 7.07\vec{j} \)
11. \( \frac{1}{5}(3\vec{i} + 4\vec{j}) = 0.6\vec{i} + 0.8\vec{j} \)
12. \( 5\cdot1 + 2\cdot(-3) = 5 - 6 = -1 \)
13. \( \vec{BC} = (2 - 1)\vec{i} + (-3)\vec{j} = \vec{i} - 3\vec{j} \)
14. \( \tan^{-1}\left(\frac{-2}{2}\right) = -45^\circ \)
15. \( \sqrt{60^2 + 20^2} = \sqrt{4000} \approx 63.25 \text{ m/s} \)
16. \( (3 -1)\vec{i} + (1 + 4)\vec{j} = 2\vec{i} + 5\vec{j} \)
17. \( -3\vec{i} + 2\vec{j} \)
18. Magnitude: \( \sqrt{5^2 + 5^2} = 7.07 \), Direction = \( 45^\circ \)
19. Midpoint = \( (3,4) \)
20. k = -3
21. \( 7\vec{i} - 3\vec{j} \)
22. Dot product = 1 + (-1) = 0 → Perpendicular
23. \( 6\vec{i} - 8\vec{j} \)
24. \( \cos\theta = \frac{6}{\sqrt{13}\cdot 2} \)
25. \( \sqrt{10^2 + 6^2} = \sqrt{136} = 11.66 \)
26. \( \vec{F}_R = 4\vec{i} + 3\vec{j}, \text{ magnitude } = 5 \)
27. \( \frac{-7\vec{i} + 24\vec{j}}{25} \)
28. Yes, same direction ratio
29. \( -6\vec{i} + 3\vec{j} \)
30. \( \vec{AB} = 4\vec{i} + 3\vec{j}, \vec{BC} = 2\vec{i} - 2\vec{j}, \vec{AC} = 6\vec{i} + \vec{j} \)