To find a vector that has the same direction as the given vector (6, 6, 2) but with a different length (specifically, a length of 6), you can follow these steps:
- Calculate the length (magnitude) of the original vector:
- The formula for the magnitude of a vector (x, y, z) is:
- Magnitude =
√(x² + y² + z²) - For the vector (6, 6, 2):
- Magnitude =
√(6² + 6² + 2²)=√(36 + 36 + 4)=√76≈8.717 - Find the unit vector:
- The unit vector in the same direction is obtained by dividing each component of the original vector by its magnitude:
- Unit vector = (6 / √76, 6 / √76, 2 / √76)
- Scale the unit vector to the desired length:
- To obtain a vector with a length of 6, multiply the unit vector by 6:
- New vector = (6 * (6 / √76), 6 * (6 / √76), 6 * (2 / √76))
- Calculating the components of the new vector:
- New vector = ((36 / √76), (36 / √76), (12 / √76))
- To further simplify, you can calculate these values approximately:
- √76 ≈ 8.717
- New vector ≈ (
4.130,4.130,1.376)
Thus, the vector that has the same direction as (6, 6, 2) but has a length of 6 is approximately (4.130, 4.130, 1.376).