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

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To find a vector with the same direction as the vector (6, 6, 6) but with a specified length of 6 units, we first need to determine the unit vector in the direction of (6, 6, 6). A unit vector has a length (or magnitude) of 1 and retains the direction of the original vector. Here are the steps to achieve this:

  1. Calculate the magnitude of the original vector: The magnitude (or length) of the vector (6, 6, 6) can be calculated using the formula:

    2. magnitude = √(x2 + y2 + z2)

    For our vector:

    magnitude = √(62 + 62 + 62) = √(36 + 36 + 36) = √(108) = 6√{3}

  2. Find the unit vector: The unit vector in the direction of (6, 6, 6) can be found by dividing each component of the vector by its magnitude:

    3. unit vector = (6/(6√{3}), 6/(6√{3}), 6/(6√{3})) = (1/√{3}, 1/√{3}, 1/√{3})

    Thus, the unit vector in the direction of (6, 6, 6) is approximately (0.577, 0.577, 0.577).

  3. Scale the unit vector to the desired length: To get a vector of length 6, multiply the unit vector by 6:

    4. new vector = 6 * (1/√{3}, 1/√{3}, 1/√{3}) = (6/√{3}, 6/√{3}, 6/√{3})

    Therefore, a vector that has the same direction as (6, 6, 6) but has a length of 6 is:

    (6/√{3}, 6/√{3}, 6/√{3})

In summary, the new vector is (6/√{3}, 6/√{3}, 6/√{3}), while retaining the same direction as the original vector.

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