To find a vector that has the same direction as the vector (2, 4, 2) and a specified length of 6, we need to follow a few steps involving vector normalization and scaling.
First, we determine the length (or magnitude) of the original vector (2, 4, 2). The magnitude is calculated using the formula:
Magnitude = √(x² + y² + z²)
Applying this to our vector:
Magnitude = √(2² + 4² + 2²) = √(4 + 16 + 4) = √24 = 2√6
Next, we need to normalize the vector to get a unit vector in the same direction. We do this by dividing each component of the vector by its magnitude:
Unit Vector = (2 / (2√6), 4 / (2√6), 2 / (2√6)) = (1/√6, 2/√6, 1/√6)
Now that we have the unit vector, we can scale it to the desired length of 6 by multiplying each component of the unit vector by 6:
Vector of length 6 = 6 × (1/√6, 2/√6, 1/√6) = (6/√6, 12/√6, 6/√6) = (√6 × 6/6, √6 × 12/6, √6 × 6/6) = (√6, 2√6, √6)
Therefore, the vector that has the same direction as (2, 4, 2) and a length of 6 is:
(√6, 2√6, √6)