Understanding Vectors

What is a Vector?

A vector is a quantity that has both magnitude (size) and direction. For example, if you say “I walked 5 metres to the north,” you have both the distance (5 metres) and the direction (north).

How to Represent Vectors

Vectors can be represented graphically as arrows. The length of the arrow shows the magnitude, and the direction of the arrow shows the direction of the vector.

Adding Vectors

When we add vectors, we are combining their magnitudes and directions to find a resultant vector.

Key Rules for Adding Vectors

  1. Graphical Method: You can add vectors by placing them head to tail. Draw the first vector, then start the next vector from the tip (head) of the previous one. The resultant vector is drawn from the tail of the first vector to the head of the last vector.
  2. Component Method: If you know the components (horizontal and vertical parts) of each vector, you can add them algebraically:
    • For vector A with components ((A_x, A_y)) and vector B with components ((B_x, B_y)), the resultant vector R will be:
    R_x = A_x + B_x
    R_y = A_y + B_y
    The resultant vector can then be expressed as ( R = (R_x, R_y) ).

Example of Adding Vectors Graphically

Imagine you have two vectors:

  • Vector A: 4 units to the right (East).
  • Vector B: 3 units up (North).

To add them graphically:

  1. Draw vector A.
  2. Starting from the end of vector A, draw vector B.
  3. Draw a line from the start of vector A to the end of vector B. This line is the resultant vector.

Example of Adding Vectors Using Components

Let’s say:

  • Vector A = (3, 2)
  • Vector B = (1, 4)

Using the component method:

  • ( R_x = 3 + 1 = 4 )
  • ( R_y = 2 + 4 = 6 )

So, the resultant vector ( R ) is (4, 6).

Tips and Tricks

  • Always pay attention to the direction: If you are moving in different directions, make sure to treat them appropriately (e.g., up is positive, down is negative).
  • Draw it out: Visualising vectors can help you understand how to add them correctly.
  • Use a ruler and protractor when drawing to ensure accuracy.

Questions

Easy Level Questions

  1. If vector A is 2 units to the right and vector B is 3 units to the right, what is the resultant vector?
  2. Add vector A (4, 0) and vector B (0, 3). What is the resultant vector?
  3. Vector A is 5 units down. Vector B is 2 units down. What is the resultant vector?
  4. If vector A is 1 unit up and vector B is 1 unit down, what is the resultant vector?
  5. Vector A is (2, 3) and vector B is (3, 2). Find the resultant vector.
  6. If vector A is 6 units left and vector B is 4 units left, what is the resultant vector?
  7. Add vector A (1, 1) and vector B (1, 1). What is the resultant vector?
  8. Vector A is 3 units to the north. Vector B is 2 units to the north. What is the resultant vector?
  9. If vector A is (5, 0) and vector B is (0, 5), what is the resultant vector?
  10. Vector A is 2 units to the east, and vector B is 2 units to the east. What is the resultant vector?

Medium Level Questions

  1. Vector A is (3, 4) and vector B is (1, 2). What is the resultant vector?
  2. If vector A is 4 units to the east and vector B is 3 units to the north, what is the resultant vector?
  3. Add vector A (5, 2) to vector B (1, 3). What is the resultant vector?
  4. Vector A is (2, 5) and vector B is (3, -1). Find the resultant vector.
  5. If vector A is (0, 4) and vector B is (3, 0), what is the resultant vector?
  6. Add vector A (4, 1) and vector B (2, 2). What is the resultant vector?
  7. If vector A is 6 units to the left and vector B is 3 units down, what is the resultant vector in component form?
  8. Vector A is 3 units to the east and vector B is 4 units to the south. What is the resultant vector?
  9. Add vector A (7, 2) and vector B (-3, 5). What is the resultant vector?
  10. If vector A is (6, 0) and vector B is (0, -3), what is the resultant vector?

Hard Level Questions

  1. If vector A = (3, 4) and vector B = (5, -1), find the resultant vector.
  2. Vector A is 2 units at an angle of 30° to the horizontal, and vector B is 4 units at an angle of 60°. Find the resultant vector using components.
  3. Vector A is (2, 3) and vector B is (-1, 4). Find the resultant vector.
  4. If vector A is 5 units to the north and vector B is 5 units to the east, what is the magnitude of the resultant vector?
  5. Add vector A (1, 1) and vector B (2, 2). What is the resultant vector?
  6. If vector A is (4, 0) and vector B is (0, 4), calculate the resultant vector.
  7. Vector A is at an angle of 45° with a magnitude of 5 units, and vector B is at an angle of 135° with a magnitude of 3 units. What is the resultant vector?
  8. Find the resultant vector for vector A = (6, 2) and vector B = (4, -3).
  9. If vector A is (3, 4) and vector B is (-3, 2), find the resultant vector.
  10. For vectors A = (2, 1) and B = (1, -1), find the resultant vector.

Answers

Easy Level Answers

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

Medium Level Answers

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

Hard Level Answers

  1. (8, 3)
  2. Use components: A = (1.732, 1) and B = (2, -2.598) Resultant = (3.732, -1.598)
  3. (1, 7)
  4. Magnitude = \sqrt{(5^2 + 5^2)} = 5\sqrt{2}
  5. (3, 3)
  6. (4, 4)
  7. (4.24, 1.24)
  8. (10, -1)
  9. (0, 6)
  10. (3, 0)

Feel free to ask questions if you’re unsure about any of the concepts or problems!