# Find the value of $m$ for which the vectors $\overrightarrow{a}=\overrightarrow{3i}+\overrightarrow{2j}+\overrightarrow{9k}$ and $\overrightarrow{b}=\overrightarrow{i}+\overrightarrow{mj}+\overrightarrow{3k}$ are parallel

Toolbox:
• If $\overrightarrow a = a_1\overrightarrow i + a_2 \overrightarrow j+a_3 \overrightarrow k,\: \: \overrightarrow b = b_1 \overrightarrow i+b_2\overrightarrow j + b_3 \overrightarrow k$ then $\overrightarrow a.\overrightarrow b = a_1b_1+a_2b_2+a_3b_3$
• If $\overrightarrow a \perp \overrightarrow b$ then $\overrightarrow a.\overrightarrow b=0$ and for nonzero vectors if $\overrightarrow a.\overrightarrow b=0 \Rightarrow \overrightarrow a \perp \overrightarrow b.$
• If $\overrightarrow a \parallel \overrightarrow b$ then $\overrightarrow a.\overrightarrow b=|\overrightarrow a|\: |\overrightarrow b|$ if they are in the same direction and $\overrightarrow a.\overrightarrow b=-|\overrightarrow a||\overrightarrow b|$ if they are in opposite directions.
• If $\overrightarrow a = a_1\overrightarrow i+a_2\overrightarrow j+a_3\overrightarrow k$ then $|\overrightarrow a|=\sqrt{a_1^2+a_2^2+a_3^3}$
Step 1
$|\overrightarrow a| = \sqrt{9+4+81}=\sqrt{94} \: |\overrightarrow b|=\sqrt{1+m^2+9}=\sqrt{10}+m^2$
If $\overrightarrow a \parallel \overrightarrow b$ and they are in the same direction, then $\overrightarrow a.\overrightarrow b=|\overrightarrow a||\overrightarrow b|$ and if they are in opposite directions then $\overrightarrow a.\overrightarrow b=-|\overrightarrow a||\overrightarrow b|$
Step 2
$\therefore (\overrightarrow a.\overrightarrow b)^2=|\overrightarrow a|^2|\overrightarrow b|^2$
$(30+2m)^2=(94)(10+m^2)$
$900+120m+4m^2=940+94m^2$
$90m^2-120m+40=0$
$\therefore 9m^2-12m+4=0$
$(3m-2)^2=0$
$m = \large\frac{2}{3}$