How can I find a vector that points in the same direction as the vector (6, 4, 2) but has a length of 6?

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To find a vector that has the same direction as the vector (6, 4, 2) but with a specific length (in this case, 6), you’ll need to follow these steps:

  1. Calculate the length (magnitude) of the original vector:

    2. The length of a vector (x, y, z) can be calculated using the formula:

    length = √(x² + y² + z²)

    For the vector (6, 4, 2), the calculation is:

    length = √(6² + 4² + 2²) = √(36 + 16 + 4) = √(56) ≈ 7.48

  2. Find the unit vector:

    3. A unit vector in the same direction as (6, 4, 2) can be found by dividing each component of the vector by its magnitude:

    unit vector = (6 / 7.48, 4 / 7.48, 2 / 7.48) ≈ (0.8038, 0.5345, 0.2673)

  3. Scale the unit vector to the desired length:

    4. To obtain a vector with length 6, multiply each component of the unit vector by 6:

    vector of length 6 = (0.8038 * 6, 0.5345 * 6, 0.2673 * 6) ≈ (4.823, 3.207, 1.603)

In summary, the vector that has the same direction as (6, 4, 2) and a length of 6 is approximately:

(4.823, 3.207, 1.603)

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