Mathematics - Vector Question with Solution | TestHub

When the two lines intersect then shortes distance between them is zero i.e.
$\begin{array}{rl}& \frac{\left({\vec{a}}_2-{\vec{a}}_1\right)\cdot {\vec{b}}_1\times {\vec{b}}_2}{\left|{\vec{b}}_1\times {\vec{b}}_2\right|}=0\\ & \Rightarrow \left({\vec{a}}_2-{\vec{a}}_1\right)\cdot {\vec{b}}_1\times {\vec{b}}_2=0\end{array}$
where ${\vec{a}}_1=\hat{i}+2\hat{j}+3\hat{k},{\vec{b}}_1=k\hat{i}+2\hat{j}+3\hat{k}$
$\begin{array}{rl}& {\vec{a}}_2=2\hat{i}+3\hat{j}+\hat{k},{\hat{b}}_2=3\hat{i}+k\hat{j}+2\hat{k}\\ & \Rightarrow \left|\begin{array}{ccc}1& 1& -2\\ k& 2& 3\\ 3& k& 2\end{array}\right|=0\\ & \Rightarrow 1(4-3k)-1(2k-9)-2\left(k^2-6\right)=0\\ & \Rightarrow -2k^2-5k+25=0\Rightarrow k=-5\text{ or }\frac{5}{2}\end{array}$