Picture it first

Drag the green and rust tips.

$\vec{a} = \langle 4, 2 \rangle$

$\vec{b} = \langle 1, 3 \rangle$

$\vec{a}+\vec{b} = \langle 5, 5 \rangle$

Magnitude $= 7.1$

Direction $= 45^\circ$

The dashed rust arrow shows b laid tip-to-tail on a; the heavy ink arrow is their sum. Components just add.

Try it. Move the vectors and watch the resultant follow.

Grade 12 · Practice problems

Vector components

50 written problems across two levels — 32 foundational · 18 advanced. Work through them on paper, then reveal every answer to check. These are not graded and are counted separately from adaptive practice.

Foundational

1. FoundationalA vector of magnitude $20$ points at $30^\circ$ above the x-axis. Find its x-component.

2. FoundationalA vector of magnitude $20$ points at $30^\circ$ above the x-axis. Find its y-component.

3. FoundationalFind the magnitude of a vector with components $(5, 12)$ .

4. FoundationalFind the direction (from the x-axis) of a vector with components $(4, 4)$ .

5. FoundationalA $150\ \text{N}$ force points at $53^\circ$ above the horizontal. Find its horizontal component.

6. FoundationalA $150\ \text{N}$ force points at $53^\circ$ above the horizontal. Find its vertical component.

7. FoundationalAdd two displacements: $30\ \text{m}$ east and $40\ \text{m}$ north. Find the resultant magnitude.

8. FoundationalFor the previous displacement, find the direction north of east.

9. FoundationalA vector has $A_x = -6$ and $A_y = 8$ . Find its magnitude.

10. FoundationalA vector points due west with magnitude $12$ . Write its components.

11. FoundationalTwo forces, $3\ \text{N}$ east and $4\ \text{N}$ west, act on a point. Find the net force.

12. FoundationalFind the dot product of $(3, 4)$ and $(4, -3)$ .

13. FoundationalA boat's velocity has components $(6.0, 8.0)\ \text{m/s}$ . Find its speed.

14. FoundationalA $200\ \text{N}$ force acts at $25^\circ$ above the horizontal. Find its vertical component.

15. FoundationalA hiker walks $5.0\ \text{km}$ east, then $5.0\ \text{km}$ north, then $5.0\ \text{km}$ west. Find the resultant displacement.

16. FoundationalTwo vectors $\vec{A}=(2,3)$ and $\vec{B}=(4,-1)$ are added. Find the resultant components.

17. FoundationalA vector of magnitude $40$ points at $60^\circ$ above the x-axis. Find its x-component.

18. FoundationalA vector of magnitude $40$ points at $60^\circ$ above the x-axis. Find its y-component.

19. FoundationalFind the magnitude of a vector with components $(9, 12)$ .

20. FoundationalFind the magnitude of a vector with components $(8, 15)$ .

21. FoundationalFind the direction (from the x-axis) of a vector with components $(3,\ 3\sqrt{3})$ .

22. FoundationalA vector points due south with magnitude $9$ . Write its components.

23. FoundationalA force of $60\ \text{N}$ acts at $45^\circ$ above the horizontal. Find its horizontal component.

24. FoundationalAdd two displacements: $9\ \text{m}$ east and $12\ \text{m}$ north. Find the resultant magnitude.

25. FoundationalFind the dot product of $(2, 5)$ and $(3, 1)$ .

26. FoundationalA vector has components $(-4,\ -3)$ . Find its magnitude.

27. FoundationalTwo forces, $7\ \text{N}$ north and $5\ \text{N}$ south, act on a point. Find the net force.

28. FoundationalA velocity has components $(12,\ 5)\ \text{m/s}$ . Find the speed.

29. FoundationalA vector of magnitude $26$ has a y-component of $10$ . Find its x-component (take it positive).

30. FoundationalTwo vectors $\vec{A}=(5,\ -2)$ and $\vec{B}=(-1,\ 6)$ are added. Find the resultant components.

31. FoundationalA $500\ \text{N}$ force acts at $20^\circ$ above the horizontal. Find its vertical component.

32. FoundationalA vector makes $70^\circ$ with the x-axis and has magnitude $12$ . Find its x-component.

Advanced

33. AdvancedA vector of magnitude $10$ has an x-component of $6$ . Find its y-component (take it positive).

34. AdvancedThree forces act on a point: $10\ \text{N}$ east, $10\ \text{N}$ north, and $10\ \text{N}$ at $45^\circ$ south of west. Find the resultant magnitude.

35. AdvancedA vector $\vec{A}$ has magnitude $50$ at $37^\circ$ above the x-axis; $\vec{B}$ has magnitude $30$ at $150^\circ$ from the x-axis. Find the magnitude of $\vec{A}+\vec{B}$ .

36. AdvancedFind the angle between vectors $\vec{A}=(1, 0)$ and $\vec{B}=(1, 1)$ using the dot product.

37. AdvancedA vector of magnitude $R$ is the sum of two equal-magnitude vectors of size $A$ separated by $120^\circ$ . Show that $R = A$ .

38. AdvancedFor what angle between two equal vectors of magnitude $A$ is the resultant also of magnitude $A$ ?

39. AdvancedTwo forces of $12\ \text{N}$ and $16\ \text{N}$ act with $120^\circ$ between them. Find the resultant magnitude.

40. AdvancedFind the angle between $\vec{A}=(3,\ 1)$ and $\vec{B}=(1,\ 2)$ using the dot product.

41. AdvancedVector $\vec{A}$ is $60$ at $25^\circ$ ; $\vec{B}$ is $40$ at $135^\circ$ (from the x-axis). Find the magnitude of $\vec{A}+\vec{B}$ .

42. AdvancedA sign of weight $200\ \text{N}$ hangs from two ropes, each at $30^\circ$ above the horizontal on opposite sides. Find the tension in each rope.

43. AdvancedA $100\ \text{N}$ box rests on a frictionless ramp inclined at $25^\circ$ . Find the component of its weight along the ramp.

44. AdvancedFor the same box on the $25^\circ$ ramp, find the component of weight perpendicular to the ramp.

45. AdvancedThree forces act on a point: $8\ \text{N}$ east, $6\ \text{N}$ north, and $10\ \text{N}$ at $37^\circ$ north of east. Find the resultant magnitude.

46. AdvancedA resultant has $R_x = -7$ and $R_y = -24$ . Find its magnitude and direction from the positive x-axis.

47. AdvancedVectors $\vec{A}=(6,\ 8)$ and $\vec{B}=(-8,\ 6)$ . Show they are perpendicular and find the magnitude of $\vec{A}-\vec{B}$ .

48. AdvancedTwo equal forces of magnitude $F$ have a resultant of magnitude $F\sqrt{3}$ . Find the angle between them.

49. AdvancedA box's weight component down a ramp is $30\ \text{N}$ and perpendicular to the ramp is $40\ \text{N}$ . Find the total weight and the ramp angle.

50. AdvancedThree equal vectors of magnitude $10$ are arranged symmetrically, each $120^\circ$ from the next. Find their resultant.