Find a Vector that has the same Direction as −6, 6, 6 but has Length 6.

We will be using the concept of scaling and the similarity of the vectors.

Answer: A vector with the same direction as (−6, 6, 6) but has length 6 is (−6/√3, 6/√3, 6/√3).

Let's solve this step by step.

Explanation:

Given:

\(A_{0}\) = (−6, 6, 6)

Let \(A_{1}\) be the vector in the same direction as \(A_{0}\) but with length 6.

Thus, \(A_{1}\) = (−6u, 6u, 6u)

The length of a vector with coordinates (−6u, 6u, 6u) is equal to √(36 u² + 36 u² + 36 u²) = u ⋅ √108 

Given that length should be 6 units.

⇒ u ⋅ √108  = 6 

u = 6/6√3 = 1/√3

\(A_{1}\) = (−6 × 1/√3, 6 × 1/√3, 6 × 1/√3)

\(A_{1}\) = (−6/√3, 6/√3, 6/√3)

Hence, a vector with the same direction as (−6, 6, 6) but has length 6 is (−6/√3, 6/√3, 6/√3).