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# 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 perpendicular

• 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 $|\overrightarrow a|=\sqrt{a_1^2+a_2^2+a_3^3}$
$\overrightarrow a.\overrightarrow b = (3)(1)+(2)(m) +(9)(3)=3+2m+27=30+2m$
If $\overrightarrow a \perp \overrightarrow b$ then $\overrightarrow a.\overrightarrow b=0.\: \: \: \therefore 30+2m=0\:\: \: 2m=-30\: \: \: m=-15$