Find a vector that has the same direction as (−4, 6, 6) but has length 6.

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

Answer: A vector that has the same direction as (−4, 6, 6) but has length 6 is (−12/√22, 18/√22, 18/√22).

Let's solve this step by step.

Explanation:

Given, \(A_{0}\) = (−4, 6, 6)

Let \(A_{1}\) be the vector in the same direction as \(A_{0}\) but with length 6: \(A_{1}\) = (−4u, 6u, 6u)

The length of a vector with coordinates (−4u, 6u, 6u) is equal to √(16 u² + 36 u² + 36 u²) = u × √88

Given that length should be 6:

⇒ u × √88  = 6

u = 6 / 2√22 = 3/√22

\(A_{1}\) = (−4 × 3/√22, 6 × 3/√22, 6 × 3/√22)

\(A_{1}\) = (−12/√22, 18/√22, 18/√22)

Hence, a vector with the same direction as (−4, 6, 6) but has length 6 is (−12/√22, 18/√22, 18/√22).