Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-zero vectors such that $\vec{b}$ and $\vec{c}$ are non-collinear. If $\vec{a}+5 \vec{b}$ is collinear with $\vec{c}, \vec{b}+6 \vec{c}$ is collinear with $\vec{a}$ and $\vec{a}+\alpha \vec{b}+\beta \vec{c}=\overrightarrow{0}$, then $\alpha+\beta$ is equal to
Solution
<p>$$\begin{aligned}
& \vec{a}+5 \vec{b}=\lambda \vec{c} \\
& \vec{b}+6 \vec{c}=\mu \vec{a}
\end{aligned}$$</p>
<p>Eliminating $\vec{a}$</p>
<p>$$\begin{aligned}
& \lambda \overrightarrow{\mathrm{c}}-5 \overrightarrow{\mathrm{b}}=\frac{6}{\mu} \overrightarrow{\mathrm{c}}+\frac{1}{\mu} \overrightarrow{\mathrm{b}} \\
& \therefore \mu=\frac{-1}{5}, \lambda=-30 \\
& \alpha=5, \beta=30
\end{aligned}$$</p>
About this question
Subject: Mathematics · Chapter: Vector Algebra · Topic: Types of Vectors
This question is part of PrepWiser's free JEE Main question bank. 169 more solved questions on Vector Algebra are available — start with the harder ones if your accuracy is >70%.