Find a Vector that has the same Direction as −2, 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 (−2, 6, 6) but has length 6 is (−12/√76, 36/√76, 36/√76).

Let's solve this step by step.

Explanation:

Given:

\(A_{0}\) = (−2, 6, 6)

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

Thus, \(A_{1}\) = (−2u, 6u, 6u)

The length of a vector with coordinates (−2u, 6u, 6u) is equal to √(4 u² + 36 u² + 36 u²) = u ⋅ √76 

Given that length should be 6.

⇒ u ⋅ √76  = 6 

u = 6/√76 

\(A_{1}\) = (−2 × 6/√76, 6 × 6/√76, 6 × 6/√76)

\(A_{1}\) = (−12/√76, 36/√76, 36/√76)

Hence, a vector with the same direction as (−2, 6, 6) but has length 6 is (−12/√76, 36/√76, 36/√76).