Find a Vector that has the same Direction as −4, 6, 4 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 (−4, 6, 4) but has length 6 is (−12/√17, 18/√17, 12/√17).

Let's solve this step by step.

Explanation:

Given:

\(A_{0}\) = (−4, 6, 4)

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

Thus, \(A_{1}\) = (−4u, 6u, 4u)

The length of a vector with coordinates (−4u, 6u, 4u) is equal to √(16 u² + 36 u² + 16 u²) = u ⋅ √68 

Given that length should be 6.

⇒ u ⋅ √68  = 6 

⇒ u = 6/√68

⇒ u = 6/2√17 = 3/√17

\(A_{1}\) = (−4 × 3/√17, 6 × 3/√17, 4 × 3/√17)

\(A_{1}\) = (−12/√17, 18/√17, 12/√17)

Hence, a vector with the same direction as (−4, 6, 4) but has length 6 is (−12/√17, 18/√17, 12/√17).