To find a vector that points in the same direction as the vector (4, 6, 4) but has a length of 6, you can follow these steps:
- Calculate the length (magnitude) of the original vector:
The magnitude of a vector (x, y, z) is given by the formula:
Magnitude = √(x² + y² + z²)
For the vector (4, 6, 4):
Magnitude = √(4² + 6² + 4²) = √(16 + 36 + 16) = √68 ≈ 8.25
- Find the unit vector:
A unit vector is a vector that has a length of 1 and retains the same direction as the original vector. To find the unit vector, you divide each component of the original vector by its magnitude:
Unit vector = (4/√68, 6/√68, 4/√68) ≈ (0.485, 0.733, 0.485)
- Scale the unit vector to the desired length:
To get a vector with a specific length, you multiply the unit vector by that length. In this case, we want a length of 6:
New vector = 6 * (0.485, 0.733, 0.485) ≈ (2.91, 4.40, 2.91)
Therefore, a vector that has the same direction as (4, 6, 4) but has a length of 6 is approximately (2.91, 4.40, 2.91).