Introduction to Vectors

Today, we will learn about vectors and how to find their magnitude. A vector is a type of quantity that has both magnitude (how much) and direction (which way). Think of a vector as an arrow: the length of the arrow shows the magnitude, and the direction the arrow points shows the direction.

What is Magnitude?

The magnitude of a vector is simply its length. To find the magnitude, we often use the Pythagorean theorem, especially when we have a vector represented in a coordinate system.

Formula for Magnitude

If we have a vector v represented as v = (x, y), where x and y are the horizontal and vertical components respectively, the magnitude |v| is given by the formula:

|v| = \sqrt{x^2 + y^2}

Step-by-Step Example

Let’s find the magnitude of a vector v = (3, 4).

  1. Identify the components: Here, x = 3 and y = 4.
  2. Substitute the values into the formula:|v| = \sqrt{3^2 + 4^2}
  3. Calculate the squares:|v| = \sqrt{9 + 16}
  4. Add the results:|v| = \sqrt{25}
  5. Finally, take the square root:|v| = 5

So, the magnitude of the vector (3, 4) is 5.

Key Rules

  • Always identify the components of the vector.
  • Use the Pythagorean theorem to calculate the magnitude.
  • Remember to square the components, add them, and then take the square root.

Tips and Tricks

  • Draw a diagram! Visualising vectors as arrows on a graph can help you understand their direction and magnitude better.
  • Practice with different vectors to build your confidence.
  • Remember that the magnitude is always a non-negative number.

Practice Questions

Easy Level Questions

  1. Find the magnitude of the vector (1, 2).
  2. Find the magnitude of the vector (0, 5).
  3. Find the magnitude of the vector (2, 0).
  4. Find the magnitude of the vector (3, 4).
  5. Find the magnitude of the vector (6, 8).
  6. Find the magnitude of the vector (1, 1).
  7. Find the magnitude of the vector (4, 3).
  8. Find the magnitude of the vector (5, 12).
  9. Find the magnitude of the vector (7, 24).
  10. Find the magnitude of the vector (0, 0).

Medium Level Questions

  1. Find the magnitude of the vector (-3, -4).
  2. Find the magnitude of the vector (10, 0).
  3. Find the magnitude of the vector (8, 15).
  4. Find the magnitude of the vector (-5, 12).
  5. Find the magnitude of the vector (9, -12).
  6. Find the magnitude of the vector (2, 2).
  7. Find the magnitude of the vector (-7, -1).
  8. Find the magnitude of the vector (4, -3).
  9. Find the magnitude of the vector (3, -4).
  10. Find the magnitude of the vector (6, -8).

Hard Level Questions

  1. Find the magnitude of the vector (5, 5).
  2. Find the magnitude of the vector (-12, -16).
  3. Find the magnitude of the vector (1, -1).
  4. Find the magnitude of the vector (9, 40).
  5. Find the magnitude of the vector (11, 60).
  6. Find the magnitude of the vector (-15, 36).
  7. Find the magnitude of the vector (30, 40).
  8. Find the magnitude of the vector (25, -7).
  9. Find the magnitude of the vector (-8, 15).
  10. Find the magnitude of the vector (14, -48).

Answers with Explanations

Easy Level Answers

  1. Magnitude = 2.236 (approx.)
  2. Magnitude = 5
  3. Magnitude = 2
  4. Magnitude = 5
  5. Magnitude = 10
  6. Magnitude = 1.414 (approx.)
  7. Magnitude = 5
  8. Magnitude = 13
  9. Magnitude = 25
  10. Magnitude = 0

Medium Level Answers

  1. Magnitude = 5
  2. Magnitude = 10
  3. Magnitude = 17
  4. Magnitude = 13
  5. Magnitude = 15
  6. Magnitude = 2.828 (approx.)
  7. Magnitude = 7.071 (approx.)
  8. Magnitude = 5
  9. Magnitude = 5
  10. Magnitude = 10

Hard Level Answers

  1. Magnitude = 7.071 (approx.)
  2. Magnitude = 20
  3. Magnitude = 1.414 (approx.)
  4. Magnitude = 41
  5. Magnitude = 61
  6. Magnitude = 39
  7. Magnitude = 50
  8. Magnitude = 26.248 (approx.)
  9. Magnitude = 17
  10. Magnitude = 50

By practicing these questions, you will become more familiar with finding the magnitude of vectors. Keep up the great work!