Let \(\vec{a}=\hat{i}+2\hat{j}+3\hat{k},\vec{b}=3\hat{i}+\hat{j}-\hat{k}\) and \(\vec{\mathrm{c}}\) be three vectors suc…

\(\text{Given, }\vec{a}=\hat{i}+2\hat{j}+3\hat{k}\&\vec{b}=3\hat{i}+\hat{j}-\hat{k}\\ \vec{a}\cdot \vec{b}=2\&\vec{a}\cdot \vec{c}=5\&\vec{b}\cdot \vec{c}=0\\ |\vec{a}|=\sqrt{14}\text{ and }|\vec{b}|=\sqrt{11}\\ \vec{c}=\lambda \vec{a}+\mu \vec{b}\\ \vec{a}\cdot \vec{c}=\lambda |\vec{a}{|}^{2}+\mu \vec{a}\cdot \vec{b}\\ 5=\lambda \times 14+\mu \times 2........\text{(i)}\\ \vec{b}\cdot \vec{c}=\lambda \vec{a}\cdot \vec{b}+\mu (\vec{b}{)}^{2}\\ 0=2\lambda +11\mu ........\text{.(ii)}\\ \text{On solving, (i) \& (ii), we get}\\ \mu =-\frac{1}{15}\lambda =\frac{11}{30}\\ \text{Now},\vec{c}=\frac{1}{6}\hat{i}+\frac{2}{3}\hat{j}+\frac{7}{6}\hat{k}\\ |\vec{c}|=\sqrt{\frac{11}{6}}\)