# If $\overrightarrow a\:and\:\overrightarrow c$ are unit and collinear vectors and $|\overrightarrow b|=6$, then $\overrightarrow b-3\overrightarrow c=\lambda \overrightarrow a,$ if $\lambda=?$

$\begin{array}{1 1} -9,3 \\ 9,3 \\ 9,-3 \\ 3,-3 \end{array}$

Given: $\overrightarrow b-3\overrightarrow c=\lambda \overrightarrow a$
$\Rightarrow\:(\overrightarrow b-3\overrightarrow c).\overrightarrow c=\lambda\overrightarrow a.\overrightarrow c$
$\Rightarrow\:\overrightarrow b.\overrightarrow c-3=\lambda$ ( since $\overrightarrow a,\overrightarrow c$ are unit collinear vectors.)
or $\overrightarrow b.\overrightarrow c=3+\lambda$
Also from given statement $\Rightarrow\: |\overrightarrow b-3\overrightarrow c|^2=\lambda^2| \overrightarrow a|^2$
$\Rightarrow\:|\overrightarrow b|^2+9|\overrightarrow c|^2-6\overrightarrow b.\overrightarrow c=\lambda^2 |\overrightarrow a|^2$
$\Rightarrow\:36+9-6(3+\lambda)=\lambda^2$
Solving which we get $\lambda = -9,3$