Year 11 Maths: Find the Component Form of a Vector

Understanding Vectors

A vector is a quantity that has both magnitude (how much) and direction (where to). You can think of a vector as an arrow pointing from one place to another.

What is Component Form?

The component form of a vector breaks it down into its horizontal (x) and vertical (y) parts. It tells us how far to move right or left (x-axis) and how far to move up or down (y-axis).

How to Find the Component Form of a Vector

To find the component form of a vector, follow these steps:

1. Identify Two Points: Let’s say we have point A (x₁, y₁) and point B (x₂, y₂).
2. Calculate the Change in x: This is done by subtracting the x-coordinates: $$\Delta x = x₂ – x₁$$
3. Calculate the Change in y: This is done by subtracting the y-coordinates: $$\Delta y = y₂ – y₁$$
4. Write in Component Form: The vector from point A to point B can be written as: $$\vec{v} = (\Delta x, \Delta y)$$

Example

Let’s say we have two points: A(2, 3) and B(5, 7).

1. Identify the points: A(2, 3), B(5, 7)
2. Calculate the change in x: $$\Delta x = 5 – 2 = 3$$
3. Calculate the change in y: $$\Delta y = 7 – 3 = 4$$
4. Write in component form: $$\vec{v} = (3, 4)$$

So, the vector from A to B is (3, 4).

Key Rules to Remember

• Always subtract the starting point from the ending point (B – A).
• The first number in the component form is how far you go left/right (x-axis).
• The second number is how far you go up/down (y-axis).

Tips and Tricks

• Draw a diagram! Visualising the points can help you understand the direction of the vector.
• Double-check your calculations: Make sure you subtract the coordinates correctly.
• Practice with different points to become confident!

Practice Questions

Easy Level Questions

1. Find the component form of the vector from A(1, 2) to B(4, 5).
2. Find the component form of the vector from A(0, 0) to B(3, 3).
3. Find the component form of the vector from A(2, 1) to B(5, 1).
4. Find the component form of the vector from A(3, 4) to B(6, 7).
5. Find the component form of the vector from A(1, 1) to B(1, 5).
6. Find the component form of the vector from A(0, 2) to B(4, 2).
7. Find the component form of the vector from A(2, 3) to B(2, 6).
8. Find the component form of the vector from A(5, 5) to B(5, 1).
9. Find the component form of the vector from A(3, 2) to B(7, 5).
10. Find the component form of the vector from A(4, 3) to B(0, 3).

Medium Level Questions

11. Find the component form of the vector from A(2, 4) to B(6, 8).
12. Find the component form of the vector from A(-1, -2) to B(2, 1).
13. Find the component form of the vector from A(5, -3) to B(5, 2).
14. Find the component form of the vector from A(1, 7) to B(4, 10).
15. Find the component form of the vector from A(3, 3) to B(3, -1).
16. Find the component form of the vector from A(-2, 3) to B(2, 7).
17. Find the component form of the vector from A(0, 0) to B(-3, -3).
18. Find the component form of the vector from A(2, 5) to B(2, 8).
19. Find the component form of the vector from A(2, -2) to B(4, 0).
20. Find the component form of the vector from A(6, 4) to B(2, 0).

Hard Level Questions

21. Find the component form of the vector from A(-5, -5) to B(0, 0).
22. Find the component form of the vector from A(3, 8) to B(-1, 4).
23. Find the component form of the vector from A(7, 3) to B(2, -1).
24. Find the component form of the vector from A(-3, -1) to B(1, 5).
25. Find the component form of the vector from A(0, 0) to B(-6, 4).
26. Find the component form of the vector from A(4, 2) to B(2, -2).
27. Find the component form of the vector from A(-1, 2) to B(3, 3).
28. Find the component form of the vector from A(5, 5) to B(-2, -2).
29. Find the component form of the vector from A(-4, -4) to B(-1, -1).
30. Find the component form of the vector from A(3, 1) to B(3, 7).

Answers to Practice Questions

Easy Level Answers

1. (3, 3)
2. (3, 3)
3. (3, 0)
4. (3, 3)
5. (0, 4)
6. (4, 0)
7. (0, 3)
8. (0, -4)
9. (4, 3)
10. (-4, 0)

Medium Level Answers

11. (4, 4)
12. (3, 3)
13. (0, 5)
14. (3, 3)
15. (0, -4)
16. (4, 4)
17. (-3, -3)
18. (0, 3)
19. (2, 2)
20. (-4, -4)

Hard Level Answers

21. (5, 5)
22. (-4, -4)
23. (-5, -4)
24. (4, 6)
25. (-6, 4)
26. (-2, -4)
27. (4, 1)
28. (-7, -7)
29. (3, 3)
30. (0, 6)

Conclusion

Finding the component form of a vector is an important skill in maths. Remember to always subtract the coordinates of the starting point from the ending point. With practice, you’ll become confident in breaking down vectors into their component form! Keep working on those practice questions!