Question

If the vectors $\overrightarrow{a}$ = 2$\hat{i}$ − 3$\hat{j}$ and $\overrightarrow{b}$ =−6$\hat{i}$ + m$\hat{j}$ are collinear, find the value of m.

Answer 1

We have, $\overrightarrow{a}=2\stackrel{⏜}{i}-3\stackrel{⏜}{j}$ and $\overrightarrow{b}=-6\stackrel{⏜}{i}+m\stackrel{⏜}{j}$
Since, the vectors are collinear. We have,
$$\begin{gathered} \overrightarrow{a}=\lambda \overrightarrow{b} \\ \Rightarrow 2\stackrel{⏜}{i}-3\stackrel{⏜}{j}=\lambda \left(-6\stackrel{⏜}{i}+m\stackrel{⏜}{j}\right) \\ \Rightarrow 2=-6\lambda \mathrm{and}-3=\lambda m \\ \Rightarrow \lambda =-\frac{1}{3}\mathrm{and}m=-\frac{3}{\lambda }=-\frac{3}{-{\displaystyle \frac{1}{3}}}=9 \end{gathered}$$
Hence, $m=9$