How can I find two unit vectors that form a 60-degree angle with the vector v = (3, 4)?

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Finding Unit Vectors at a 60-Degree Angle

To find two unit vectors that form a 60-degree angle with the vector v = (3, 4), we can follow these steps:

  1. Normalize the Vector: First, we need to convert vector v into a unit vector by calculating its magnitude. This is done using the formula:

    2. Magnitude of v = √(3² + 4²) = √(9 + 16) = √25 = 5

  2. Next, we normalize v:

    3. Unit vector of v = (3/5, 4/5)

  3. Use the Angle Formula: To find the unit vectors that make a 60-degree angle with the unit vector of v, we use the rotation formula. The two unit vectors can be found by rotating the original unit vector v’ by ±60 degrees:

    4. Let:

    • u1 = cos(60°) * v’ + sin(60°) * (0, 1)
    • u2 = cos(-60°) * v’ + sin(-60°) * (0, 1)
  4. Substituting the values:
    • cos(60°) = 1/2, sin(60°) = √3/2
    • First unit vector:
      • u1 = (1/2)(3/5) + (√3/2)(4/5) = (3/10) + (4√3/10) = (3 + 4√3)/10
    • Second unit vector:
      • u2 = (1/2)(3/5) + (-√3/2)(4/5) = (3/10) - (4√3/10) = (3 - 4√3)/10

Final Unit Vectors:

After solving, we conclude:

  • Unit Vector 1: u1 = ((3 + 4√3)/10, (4 + 3√3)/10)
  • Unit Vector 2: u2 = ((3 – 4√3)/10, (4 – 3√3)/10)

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