Question

The cartesian equation of a line is $\frac{x-3}{2}$=$\frac{y+1}{-2}$=$\frac{z-3}{5}$, find the vector equation of line.

Answer 1

Cartesian equation $\Rightarrow \frac{x-3}{2}=\frac{y+1}{-2}=\frac{z-3}{5}=k\left(const\right)$

$\Rightarrow x=2k+3,y=-2k-1,z=5k+3$

Vector equation $\Rightarrow \overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}$

$\Rightarrow \overrightarrow{r}=(2k+3)\hat{i}+(-2k-1)\hat{j}+(5k+3)\hat{k}=3\hat{i}-\hat{j}+3\hat{k}+k(2\hat{i}-2\hat{j}+5\hat{k})$