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Picture it first

Drag the blue and green 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 green arrow shows b laid tip-to-tail on a; the amber arrow is their sum. Components just add.

Try it  Move the vectors and watch the resultant follow.

Grade 12 · Practice problems

Vector components

50 problems, including 18 🔥 challenging (exam-level, multi-step). Work through them on paper, then reveal every answer to check.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    33. 🔥 Challenging problems

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

  34. 34. 🔥 ChallengingThree 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. 35. 🔥 ChallengingA 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. 36. 🔥 ChallengingFind the angle between vectors \vec{A}=(1, 0) and \vec{B}=(1, 1) using the dot product.

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

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

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

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

  41. 41. 🔥 ChallengingVector \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. 42. 🔥 ChallengingA 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. 43. 🔥 ChallengingA 100\ \text{N} box rests on a frictionless ramp inclined at 25^\circ. Find the component of its weight along the ramp.

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

  45. 45. 🔥 ChallengingThree 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. 46. 🔥 ChallengingA resultant has R_x = -7 and R_y = -24. Find its magnitude and direction from the positive x-axis.

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

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

  49. 49. 🔥 ChallengingA 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. 50. 🔥 ChallengingThree equal vectors of magnitude 10 are arranged symmetrically, each 120^\circ from the next. Find their resultant.