Question

Write the vector equation of the line equation
$\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$

Answer 1

If a line passes through a point $\overrightarrow{a}$ and parallel to the vector $\overrightarrow{r}$ then the vector form of the line can be written as $\overrightarrow{a}+\lambda \left(\overrightarrow{r}\right)$, where $\lambda$ is a constant

The directional cosines of the given line are $2,3,4$

Hence, $2\hat{i}+3\hat{j}+4\hat{k}$ is parallel to the given line

The given line passes through $\hat{i}+2\hat{j}+3\hat{k}$

Hence, the vector form of the line is given by $(\hat{i}+2\hat{j}+3\hat{k})+\lambda (2\hat{i}+3\hat{j}+4\hat{k})$