Question

Let $\overline{a},\overline{b},\overline{c}$ be three non-zero vectors, no two of which are collinear. lf the vector $\overline{a}+2\overline{b}$ is collinear with $\overline{c}$ and $\overline{b}+3\overline{c}$ is collinear with $\overline{a}$ , then $\overline{a}+\overline{b}+3\overline{c}=$

• A. $\lambda \overline{a}$
• B. $\lambda \overline{b}$
• C. $\lambda \overline{c}$
• D. $0$

Answer 1

The correct option is D $0$

$\overline{a},\overline{b},\overline{c}$

$(\overline{a}+2\overline{b})$is collinear with $\overline{c}$

$(\overline{b}+3\overline{c})$ is collinear with $\overline{a}$

$\overline{a}+2\overline{b}=\lambda \overline{c}\to \left(1\right)$

$\overline{b}+3\overline{c}=\mu \overline{a}\to \left(2\right)$

$b=\mu \overline{a}-\overline{c}$

$\overline{a}+2(\mu \overline{a}-3\overline{c})=\lambda \overline{c}$

$\overline{a}-6\overline{c}=\lambda \overline{c}-2\mu \overline{a}\to$ (3)

from(1), (2) and (3)

$\overline{a}+\overline{b}+3\overline{c}=0$

OPTION : D