Find a vector that has the same direction as 6, 4, 6 but has length 6

To find a vector that has the same direction as a given vector but a different length, we can use the concept of unit vectors. A unit vector has a length of 1 and maintains the direction of the original vector.

First, we need to determine the length (magnitude) of the original vector (6, 4, 6). The magnitude is calculated using the formula:

|v| = √(x² + y² + z²)

Substituting the values:

|v| = √(6² + 4² + 6²) = √(36 + 16 + 36) = √88

Next, we find the unit vector by dividing each component of the original vector by its magnitude:

u = (6/√88, 4/√88, 6/√88)

This simplifies to:

u ≈ (0.639, 0.425, 0.639)

Now, to create a new vector that keeps the same direction but has a length of 6, we multiply the unit vector by 6:

new_vector = 6 * u ≈ 6 * (0.639, 0.425, 0.639) = (3.834, 2.55, 3.834)

Thus, the vector that has the same direction as (6, 4, 6) but a length of 6 is approximately (3.834, 2.55, 3.834).