Introductory Vectors?

Question

Please help me with the following question: Find a vector that has the same direction as <-2,4,2> but has length 6.

The answer is: <-sqrt(6)i + 2*sqrt(6)j + sqrt(6)k>. A breif idea of how to do this problem would be much appreciated.

Answer 1

The length of the original vector is sqrt [ (-2)^2 + (4)^2 + (2)^2 ] = sqrt(24) = 2 * sqrt(6).

To get a vector of length 6 in the same direction, you need to multiply the original vector by scalar x as follows. Need x such that

6 = 2sqrt(6) * x

x = 6 / 2sqrt(6) = sqrt(6) / 2.

Multiplying the original vector by this scalar gives the answer stated.