Find a vector that has the same direction as (−4, 6, 2) 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, 2) but has length 6 is (−12/√14, 18/√14, 6/√14).

Let's solve this step by step.

Explanation:

Given,  \(A_{0}\) = (−4, 6, 2).

Let  \(A_{1}\) be the vector in the same direction as  \(A_{0}\) but with length 6: \(A_{1}\) = (−4u, 6u, 2u).

The length of a vector with coordinates (−4u, 6u, 2u) is equal to √(16 u² + 36 u² + 4 u²) = u ⋅ √56

Given that length should be 6.

⇒ u ⋅ √56  = 6 

u = 6 / 2√14 = 3√14

\(A_{1}\) = (−4 × 3√14, 6 × 3√14, 2 × 3√14)

\(A_{1}\) = (−12/√14, 18/√14, 6/√14)

Hence, a vector with the same direction as (−4, 6, 2) but has length 6 is (−12/√14, 18/√14, 6/√14).