Show that if there exist scalars x and y, not both zero, such that xa+xb=0, then a and b are parallel.?

Answer 1

I think you meant xa+yb=0.
We can assume y .ne. 0, since x and y are not both 0.
Then b = -(x/y) a.
If x is 0 then b = 0, so let's assume x .ne. 0.
Then a is a scalar multiple of b, which means
a and b are parallel.