Find your Math Personality!
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Find a vector that has the same Direction as (−6, 6, 2) 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 (−6, 6, 2) but has length 6 is (−18/√19, 18/√19, 6/√19).
Let's solve this step by step.
Explanation:
Given, \(A_{0}\) = (−6, 6, 2)
Let \(A_{1}\) be the vector in the same direction as \(A_{0}\) but with length 6: \(A_{1}\) = (−6u, 6u, 2u)
The length of a vector with coordinates (−6u, 6u, 2u) is equal to √(36 u2 + 36 u2 + 4 u2) = u ⋅ √76
Given that length should be 6.
⇒ u ⋅ √76 = 6
u = 6 / 2√19 = 3/√19
\(A_{1}\) = (−6 × 3/√19, 6 × 3/√19, 2 × 3/√19)
\(A_{1}\) = (−18/√19, 18/√19, 6/√19)
Hence, a vector that has the same direction as (−6, 6, 2) but has length 6 is (−18/√19, 18/√19, 6/√19).
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