How to find a vector A that has the same direction as ⟨−8,7,8⟩ but has length 3 ?

The idea is based on a concept of scaling and similarity.
Any vector that "has the same direction" as #(-8,7,8)# has all the coordinates proportional to this given vector and, therefore, can be described by coordinates #(-8f,7f,8f)# where #f# is a scaling factor.

All we need now is to find a scaling factor that leads to a vector with the length #3#.

The length of a vector with coordinates #(-8f,7f,8f)# uguale a
#sqrt(64f^2+49f^2+64f^2) = f*sqrt(177)#

So, if we want the length to be equal to #3#, we should choose
#f = 3/sqrt(177)#

The coordinates of a vector with the same direction as #(-8,7,8)# but with the length #3# sarà
#(-24/sqrt(177),21/sqrt(177),24/sqrt(177))#