Question

Let $\vec{a},\,\vec{b},\,\vec{c}$ be three non-zero vectors which are pairwise non-collinear. If $\vec{a}+3\vec{b}$ is collinear with $\vec{c}$ and $\vec{b}+2\vec{c}$ is collinear with $\vec{a}+3\vec{b}+\vec{c}$ is :

• A. $\vec{a}$
• B. $\vec{c}$
• C. $\vec{0}$
• D. $\vec{a} + \vec{c}$

Answer 1

Answer: (C)

$\vec{0}$

Answer 2

$\vec{a} +3\vec{b} = \lambda\vec{c}\quad........\left(1\right)$ $\vec{b} + 2\vec{c} = \mu\vec{a} \quad.........\left(2\right)$ $\left(1\right) - 3\left(2\right)$ gives $\left(1 + 3?\right) \vec{a} - \left(\lambda + 6\right)\vec{c} = 0$ As $\vec{a}$ and $\vec{b}$ are non collinear $\therefore 1 + 3? = 0$ and $\lambda + 6 = 0$ From $\left(1\right) \vec{a} + 3\vec{b} + 6\vec{c} = \vec{0}$