Question

$\overline{a},\overline{b},\overline{c}$ are three non-zero vectors no two of which are collinear and the vector $\overline{a}+\overline{b}$ is collinear with $\overline{c}$, $\overline{b}+\overline{c}$ is collinear with $\overline{a}$, then $\overline{a}+\overline{b}+\overline{c}$ is equal to

• A. $a$
• B. $b$
• C. $c$
• D. None of these

Answer 1

The correct option is D None of these

Given $\overline{a},\overline{b},\overline{c}$ are three non-zero vectors no two of which are collinear.
And the vector $\overline{a}+\overline{b}$ is collinear with $\overline{c}$, $\overline{b}+\overline{c}$ is collinear with $\overline{a}$.
$\Rightarrow \overline{a}+\overline{b}=t\overline{c}$ -----(1)
$\Rightarrow \overline{b}+\overline{c}=s\overline{a}\Rightarrow \overline{c}=s\overline{a}-\overline{b}$ ----(2)
From (1) and (2), we get
$\overline{a}+\overline{b}=t(s\overline{a}-\overline{b})$
Comparing on both sides gives,
$ts=1$ and $t=-1$
$\Rightarrow t=-1,s=-1$
$\Rightarrow \overline{a}+\overline{b}+\overline{c}=0$
Hence, option D.