How do you calculate the magnitude of a vector with three components?

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To find the magnitude of a vector with three components (often denoted as v = (x, y, z)), you can use the formula derived from the Pythagorean theorem extended into three dimensions. The formula is as follows:

Magnitude of the vector (|v|) = √(x² + y² + z²)

Here’s a step-by-step approach on how to visualize and calculate it:

  1. Identify the Components: Start with your vector given in the form of its components. For example, let’s say we have a vector v = (3, 4, 5).
  2. Square Each Component: Calculate the square of each of the vector’s components:
    • x² = 3² = 9
    • y² = 4² = 16
    • z² = 5² = 25
  3. Add the Squared Components: Now, add the squared components together:
    9 + 16 + 25 = 50
  4. Take the Square Root: Finally, take the square root of the sum to find the magnitude:
    √50 ≈ 7.07

So, the magnitude of the vector v = (3, 4, 5) is approximately 7.07.

This method can be applied to any three-component vector, making it a fundamental technique in vector analysis.

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