দেওয়া আছে, $$ \begin{aligned} \vec{A}&=3 \hat{i}-2 \hat{j}+\hat{k} \\ \vec{B}&=2 \hat{i}-4 \hat{j}-3 \hat{k} \\ \vec{C}&=-\hat{i}+2 \hat{j}+2 \hat{k} \end{aligned} $$ (i) $\vec{C}$ ভেক্টরের মান নির্ণয়: $$ \begin{aligned} |\vec{C}|&=\sqrt{(-1)^2+(2)^2+(2)^2} \\ &=\sqrt{1+4+4} \\ &=\sqrt{9} \\ &=3 \end{aligned} $$ (ii) $\vec{A}+\vec{B}+\vec{C}$ ভেক্টরের মান নির্ণয়: $$ \begin{aligned} \vec{A}+\vec{B}+\vec{C}&=(3+2-1)\hat{i}+(-2-4+2)\hat{j}+(1-3+2)\hat{k} \\ &=4\hat{i}-4\hat{j}+0\hat{k} \\ &=4\hat{i}-4\hat{j} \end{aligned} $$ অতএব, $$ \begin{aligned} |\vec{A}+\vec{B}+\vec{C}|&=\sqrt{(4)^2+(-4)^2} \\ &=\sqrt{16+16} \\ &=\sqrt{32} \\ &=4\sqrt{2} \end{aligned} $$ (iii) $2\vec{A}-3\vec{B}-5\vec{C}$ ভেক্টরের মান নির্ণয়: $$ \begin{aligned} 2\vec{A}&=2(3\hat{i}-2\hat{j}+\hat{k})=6\hat{i}-4\hat{j}+2\hat{k} \\ 3\vec{B}&=3(2\hat{i}-4\hat{j}-3\hat{k})=6\hat{i}-12\hat{j}-9\hat{k} \\ 5\vec{C}&=5(-\hat{i}+2\hat{j}+2\hat{k})=-5\hat{i}+10\hat{j}+10\hat{k} \end{aligned} $$ এখন, $$ \begin{aligned} 2\vec{A}-3\vec{B}-5\vec{C}&=(6\hat{i}-4\hat{j}+2\hat{k})-(6\hat{i}-12\hat{j}-9\hat{k})-(-5\hat{i}+10\hat{j}+10\hat{k}) \\ &=(6-6+5)\hat{i}+(-4+12-10)\hat{j}+(2+9-10)\hat{k} \\ &=5\hat{i}-2\hat{j}+\hat{k} \end{aligned} $$ অতএব, $$ \begin{aligned} |2\vec{A}-3\vec{B}-5\vec{C}|&=\sqrt{(5)^2+(-2)^2+(1)^2} \\ &=\sqrt{25+4+1} \\ &=\sqrt{30} \end{aligned} $$