Mathematics - Vector Question with Solution | TestHub

$\begin{array}{rl}& \vec{a}\times \vec{b}+\vec{a}\times \vec{c}+\vec{b}\times \vec{c}=(1,8,13)\\ & \vec{a}\times (\vec{a}\times \vec{b})+\vec{a}\times (\vec{a}\times \vec{c})+\vec{a}\times (\vec{b}\times \vec{c})\\ & =\vec{a}\times (\hat{i}+8\hat{j}+13\hat{k})\end{array}$
$\begin{array}{rl}& (\vec{a}\cdot \vec{b})\vec{a}-a^2\vec{b}+(\vec{a}\cdot \vec{c})\vec{a}-a^2\vec{c}+(\vec{a}\cdot \vec{c})\vec{b}-(\vec{a}\cdot \vec{b})\vec{c}=\vec{a}\times (\hat{i}+8\hat{j}+13\hat{k})\\ & \Rightarrow -26\vec{a}-29\vec{b}+13\vec{a}-29\vec{c}+13\vec{b}+26\vec{c}=\vec{a}\times (\hat{i}+8\hat{j}+13\hat{k})\end{array}$ $\begin{array}{rl}& \Rightarrow -13\overrightarrow{\mathrm{a}}-16\overrightarrow{\mathrm{b}}-3\overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{a}}\times (\hat{\mathrm{i}}+8\hat{\mathrm{j}}+13\hat{\mathrm{k}})\\ & \Rightarrow -13\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}}-16{\mathrm{b}}^2-3\overrightarrow{\mathrm{b}}\cdot \overrightarrow{\mathrm{c}}=\{\overrightarrow{\mathrm{a}}\times (\hat{\mathrm{i}}+8\hat{\mathrm{j}}+13\hat{\mathrm{k}})\}\cdot \overrightarrow{\mathrm{b}}\\ & \Rightarrow (-13)(-26)-16(50)-3\overrightarrow{\mathrm{b}}\cdot \overrightarrow{\mathrm{c}}=\left|\begin{array}{ccc}2& -3& 4\\ 1& 8& 13\\ 3& 4& -5\end{array}\right|\\ & \Rightarrow -462-3\overrightarrow{\mathrm{b}}\cdot \overrightarrow{\mathrm{c}}=-396\\ & \Rightarrow \overrightarrow{\mathrm{b}}\cdot \overrightarrow{\mathrm{c}}=-22\end{array}$
Hence $24-\vec{b}\cdot \vec{c}=46$