I have a question on the following:The line r = i + ( 1 +...

$\mathbf{r}=\hat{i}+\hat{j}-3\hat{k}+t(3\hat{j}+4\hat{k}),t\in \mathbb{R}$ represents the equation of a straight line that passes through the point with position vector $\hat{i}+\hat{j}-3\hat{k}$ (take t=0). Take any two distinct points on the line, say for $t_1\ne t_2\in \mathbb{R}$
$\begin{array}{r}{\mathbf{r}}_{\mathbf{1}}=\hat{i}+\hat{j}-3\hat{k}+t_1(3\hat{j}+4\hat{k})\\ {\mathbf{r}}_{\mathbf{2}}=\hat{i}+\hat{j}-3\hat{k}+t_2(3\hat{j}+4\hat{k})\end{array}$
Then ${\mathbf{r}}_{\mathbf{1}}\mathbf{-}{\mathbf{r}}_{\mathbf{2}}$ is a vector that is parallel to the line, i.e. the line is parallel to $(t_1-t_2)(3\hat{j}+4\hat{k})$ or put simply, the line is parallel to $3\hat{j}+4\hat{k}$
The information about a vector that is parallel to a line is as important as the information about the normal to a plane. It comes in handy in almost all applications involving lines, such as finding the distance of a point from the line, finding the image of a point with respect to a line, finding the angle between a line and a plane, etc..