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

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

Answer: A vector with the same direction as (−6, 4, 6) but has length 6 is (−18/√22, 12/√22, 18/√22).

Let's understand the solution in detail.

Explanation:

Given:

A₀ = (−6, 4, 6).

Let A₁ be the vector in the same direction as A₀ but with length 6: 

Thus, A₁ = (−6u, 4u, 6u)

The length of a vector with coordinates (−6u, 4u, 6u) is  equal to √(36u² + 16u² + 36u²) = u⋅√88 

Given that length should be 6:

⇒ u⋅√88  = 6 

⇒ u = 6 / √88

⇒ u = 6 / 2√22

⇒ u = 3/√22

Substituting the value of 'u' back to A₁ we get,

A₁ = (−6 × 3/√22, 4 × 3/√22, 6 × 3/√22)

A₁ = (−18/√22, 12/√22, 18/√22)

Hence, a vector with the same direction as (−6, 4, 6) but having a length 6 is (−18/√22, 12/√22, 18/√22).