Question

The Cartesian equation of a line is $\frac{x-6}{2}=\frac{y+4}{7}=\frac{z-5}{3}$, find its vector equation.

Answer 1

Equation of line $\frac{x-6}{2}=\frac{y+4}{7}=\frac{z-5}{3}$
$\therefore$ The line is passing through the point $(6,-4,5)$ and having direction ratios $2,7,3$.
so $\overrightarrow{a}=6\hat{i}-4\hat{j}+5\hat{k}$
and $\overrightarrow{b}=2\hat{i}+7\hat{j}+3\hat{k}$
$\therefore$ The vector equation of line is $\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b}$
$\overrightarrow{r}=(6\hat{i}-4\hat{j}+5\hat{k})+\lambda (2\hat{i}+7\hat{j}+3\hat{k})$