Question

The vector and the cartesian equations of the line through the point $(5,–2,4)$ and which is parallel to the vector $2\hat{i}-\hat{j}+3\hat{k}$ are __________

Answer 1

We have, $\overrightarrow{a}=5\hat{i}-2\hat{j}+4\hat{k}$ and $\overrightarrow{b}=2\hat{i}-\hat{j}+3\hat{k}$

Therefore, the vector equation of the line is given by,
$\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b}$
$=5\hat{i}-2\hat{j}+4\hat{k}+\lambda (2\hat{i}-\hat{j}+3\hat{k})$
$=(5+2\lambda )\hat{i}-(2+\lambda )\hat{j}+(4+3\lambda )\hat{k}$

The cartesian equation of the line is given by,
$\frac{x-5}{2}=\frac{y+2}{-1}=\frac{z-4}{3}$