The vector equation of the line $\frac{{x - 5}}{3} = \frac{{y + 4}}{7} = \frac{{z - 6}}{2}$ is ………...

It is given that, $\overrightarrow {\rm{a}} = 5\widehat {\rm{i}} - 4\widehat {\rm{j}} + 6\widehat {\rm{k}}$ and

$\overrightarrow {\rm{b}} = 3\widehat {\rm{i}} + 7\widehat {\rm{j}} + 2\widehat {\rm{k}}$

So, the vector equation will be

$\overrightarrow {\rm{r}} = (5\widehat {\rm{i}} - 4\widehat {\rm{j}} + 6\widehat {\rm{k}}) + \lambda (3\widehat {\rm{i}} + 7\widehat {\rm{j}} + 2\widehat {\rm{k}}$

$\Rightarrow$ $(x\widehat {\rm{i}} + y\widehat {\rm{j}} + z\widehat {\rm{k}}) - (5\widehat {\rm{i}} - 4\widehat {\rm{j}} + 6\widehat {\rm{k}})$

$= \lambda (\widehat {\rm{i}} + 7\widehat {\rm{j}} + 2\widehat {\rm{k}})$

$\Rightarrow$ $(x - 5)\widehat {\rm{i}} + (y + 4)\widehat {\rm{j}} + (z - 6)\widehat {\rm{k}} = \lambda (3\widehat {\rm{i}} + 7\widehat {\rm{j}} + 2\widehat {\rm{k}})$