Vectors and Length

Question

Find the vector that satisfies these conditions:

a) The vector in the opposite direction to u = <5,5> and of half its length is:

b) The vector of length 10 and in the same direction as v = <-5,1,1> is:

If you could explain how you solve each step, that would be very helpful.

Answer 1

<-2.5,-2.5> has half the magnitude of <5,5>

2.5 = 5/2

and the sign change reverses its direction

<-5,1,1> has length = sqr(5^2+1^2+1^2) = sqr(27).

to lengthen it to 10, multiply by 10/sqr27

<-50/sqr27, 10/sqr27, 10/sqr27>