(a) The scalar product of the vector a = i - j +2 k with a unit vector along sum of vectors b =2 i -4 j +5 k and c =λ i -2 j -3 k is equal to 1. Find…

The scalar product of the vector $\displaystyle \vec{a}=\hat{i}-\hat{j}+2 \hat{k}$ with a unit vector along sum of vectors $\displaystyle \vec{b}=2 \hat{i}-4 \hat{j}+5 \hat{k}$ and $\displaystyle \vec{c}=\lambda \hat{i}-2 \hat{j}-3 \hat{k}$ is equal to $\displaystyle 1$ . Find the value of $\displaystyle \lambda$.

Find the shortest distance between the lines: \[\begin{aligned} & \vec{r}=(2 \hat{i}-\hat{j}+3 \hat{k})+\lambda(\hat{i}-2 \hat{j}+3 \hat{k}) \\ & \vec{r}=(\hat{i}+4 \hat{k})+\mu(3 \hat{i}-6 \hat{j}+9 \hat{k}) \end{aligned} \]