Question

If the cartesian form of a line is $\frac{3-x}{5}=\frac{y+4}{7}=\frac{2z-6}{4}$, write the vector equation for the line.

Answer 1

Given,$\frac{3-x}{5}=\frac{y+4}{7}=\frac{2z-6}{4}$

$\Rightarrow \frac{x-3}{-5}=\frac{y-(-4)}{7}=\frac{2(z-3)}{4}$
$\Rightarrow \frac{x-3}{-5}=\frac{y-(-4)}{7}=\frac{z-3}{2}$
The given line passed through A(3,-4,3) and parallel to line with direction ratios <-5,7,2>

$\therefore \text{Vector equation of line is}\overrightarrow{r}=(3\hat{i}-4\hat{j}+3\hat{k})+\lambda (-5\hat{i}+7\hat{j}+2\hat{k})$