Question

The Cartesian equation of a line is: $2x-3=3y+1=5-6z$. Find the vector equation of a line passing through $(7,-5,0)$ and parallel to the given line.

Answer 1

Given line is $L:\frac{x-3/2}{1/2}=\frac{y+1/3}{1/3}=\frac{z-5/6}{-1/6}$

That means $\overrightarrow{v}=(1/2,1/3,-1/6)$

$\therefore$ Vector equation of a line passing through $\overrightarrow{r_0}=P_0(7,-5,0)$ and parallel to $L($ Means parallel to vector $\overrightarrow{v})$

$\to L_1:\overrightarrow{r}=\overrightarrow{r_0}+\lambda \overrightarrow{v}$

$\overrightarrow{r}=(7,-5,0)+\lambda (1/2,1/3,-1/6)$