Question

The cartesian equation of a line is $\frac{x+3}{2}=\frac{y-5}{4}=\frac{z+6}{2}$ find the vector equation of the line?

Answer 1

Consider the Cartesian equation,

$\frac{x+3}{2}=\frac{y-5}{4}=\frac{z+6}{2}$

Now,

$\overrightarrow{a}=-3\hat{i}+5\hat{j}-6\hat{k}$

$\overrightarrow{b}=2\hat{i}+4\hat{j}+2\hat{k}$

Vector equation,

$\overrightarrow{r}=\overrightarrow{a}+\lambda .\overrightarrow{b}$

$=-3\hat{i}+5\hat{j}-6\hat{k}+\lambda (2\hat{i}+4\hat{j}+2\hat{k})$

This is the required equation.