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A day full of math games & activities. Find one near you.

A day full of math games & activities. Find one near you.

# 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 u^{2} + 36 u^{2} + 16 u^{2}) = 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).

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