multiplying term on sum

Question

Say I know the following relation holds

$$ \sum_i f_i + \sum_i g_i = 0 $$

Now I multipy both sides with a set of vectors $\mathbf v_i$. Will it still be true that

$$ \sum_i f_i \mathbf v_i + \sum_i g_i \mathbf v_i= \mathbf 0 $$

?

Answer 1

Not at all. This would also be invalid with scalars $v_i$.

Answer 2

NO. Let $f_1=1$ and $f_2=3$; $g_1=-2$ and $g_2=-2$. Then is $1e_1+3e_2+-2e_1+-2e_2=0$? with $e_1$, $e_2$ being perpendicular unit vectors.

Answer 3

No. Suppose $f_1=1,f_2=2,g_1=-2,g_2=-1,v_1=1,v_2=2.$ Then $$\sum_if_i+\sum_ig_i=1+2-2-1=0\\ \sum_if_iv_i+\sum_ig_iv_i=1+4-2-2=1$$