To find a vector that has the same direction as another vector but with a different magnitude, we follow these steps:
- Identify the original vector: Here, we are given the vector (2, 4, 2).
- Calculate the magnitude of the original vector: The magnitude (length) of a vector is calculated using the formula:
Magnitude = √(x² + y² + z²)
For our vector (2, 4, 2), the magnitude can be calculated as:
Magnitude = √(2² + 4² + 2²) = √(4 + 16 + 4) = √24 = 2√6
- Calculate the unit vector: To get a vector of length 1 that points in the same direction, we divide each component of the original vector by its magnitude:
Unit vector = (2, 4, 2) / (2√6) = (2/(2√6), 4/(2√6), 2/(2√6)) = (1/√6, 2/√6, 1/√6)
- Scale the unit vector to the desired length: To find the vector of length 6, we multiply the unit vector by 6:
New vector = 6 * (1/√6, 2/√6, 1/√6) = (6/√6, 12/√6, 6/√6) = (√6 * 6/6, 2√6 * 6/6, √6 * 6/6) = (√6, 2√6, √6)
Thus, the final vector that has the same direction as (2, 4, 2) but with a length of 6 is (√6, 2√6, √6).