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

Vectors are very interesting concepts in mathematics that have an immense number of applications in engineering and physics.

Answer: A vector with the same direction as (−4, 4, 6) having Length as 6 is (−12/√17, 12/√17, 18/√17).

Let's understand the solution in detail.

Explanation:

Given: Let A₀ = (−4, 4, 6).

Let A₁ be the vector in the same direction as A₀ but with a length of 6 as shown below

⇒ A₁ = (−4u, 4u, 6u)

The length of a vector with coordinates (−4u, 4u, 6u) is  equal to

√(16u² + 16u² + 36u²) = u√68

Given that the length of this vector should be 6

Thus, 

⇒ u√68  = 6 

⇒ u = 6 / √68

⇒ u = 6 / 2√17

⇒ u = 3/√17

Substituting the value of 'u' back to A₁ = (−4u, 4u, 6u)

A₁ = (−4 × 3/√17, 4 × 3/√17, 6 × 3/√17)

A₁ = (−12/√17, 12/√17, 18/√17)

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