question_answer

Find all the vectors of magnitude \[10\sqrt{3}\] that are perpendicular to the plane of \[\hat{i}+2\hat{j}+\hat{k}\] and \[-\,\hat{i}+3\hat{j}+4\hat{k}.\]

Answer:

Let        \[{{\vec{b}}_{1}}=\hat{i}+2\hat{j}+\hat{k}\] and \[{{\vec{b}}_{2}}=-\,\hat{i}+3\hat{j}+4\hat{k}\] Then, \[{{\vec{b}}_{1}}\times {{\vec{b}}_{2}}=\left| \begin{matrix}    {\hat{i}} & {\hat{j}} & {\hat{k}}  \\    1 & 2 & 1  \\    -\,1 & 3 & 4  \\ \end{matrix} \right|\]         \[=\hat{i}(5)-\hat{j}(5)+\hat{k}(5)=\vec{c}\,(say)\] \[\Rightarrow \]   \[|{{\vec{b}}_{1}}\times {{\vec{b}}_{2}}|\,\,=\sqrt{{{(5)}^{2}}+{{(5)}^{2}}+{{(5)}^{2}}}=5\sqrt{3}\] Now,     \[\hat{c}=\frac{{{{\vec{b}}}_{1}}\times {{{\vec{b}}}_{2}}}{|{{{\vec{b}}}_{1}}\times {{{\vec{b}}}_{2}}|}=\frac{{\hat{i}}}{\sqrt{3}}-\frac{{\hat{j}}}{\sqrt{3}}+\frac{{\hat{k}}}{\sqrt{3}}\] Hence, required vector \[=\pm \,(10\sqrt{3}\hat{c})\] \[=\pm \,(10\hat{i}-10\hat{j}+10\hat{k})\]