To find a vector that has the same direction as the vector (4, 4, 6) but a different length, we need to perform a series of steps that involve normalization and scaling.
Step 1: Calculate the length (magnitude) of the original vector
The length of a vector (x, y, z) is calculated using the formula:
Magnitude = √(x² + y² + z²)
For the vector (4, 4, 6):
Magnitude = √(4² + 4² + 6²)
= √(16 + 16 + 36)
= √(68)
= 2√(17)
Step 2: Normalize the vector
Normalization involves dividing each component of the vector by its magnitude. This converts the vector into a unit vector that only retains the direction:
Normalized vector = (4 / √68, 4 / √68, 6 / √68)
= (4 / (2√17), 4 / (2√17), 6 / (2√17))
= (2 / √17, 2 / √17, 3 / √17)
Step 3: Scale the unit vector to the desired length
To obtain a vector of length 6 in the same direction, we will multiply the normalized vector by 6:
Resulting vector = 6 * (2 / √17, 2 / √17, 3 / √17)
= (12 / √17, 12 / √17, 18 / √17)
Final Result
The vector that points in the same direction as (4, 4, 6) but has a length of 6 is:
(12 / √17, 12 / √17, 18 / √17)