Question

To find the vector and the Cartesian equation in symmetric form of line passing through the points, $(2,0,-3)$ and $(7,3,-10)$.

Answer 1

The required is passing through the points $(2,0,-3)$ and $(7,3,-10)$

So, it's direction ratios are proportional to $(7-2),(3-0),(-10+3)$

i.e. $5,3,-7$

therefore,

Vector equation of the line is

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

$=(2\hat{i}+0\hat{j}+3\hat{k})+\lambda (5\hat{i}+3\hat{j}-7\hat{k})$

$=(2\hat{i}+3\hat{k})+\lambda (5\hat{i}+3\hat{j}-7\hat{k})$

And, Symmetric equation is

$\begin{array}{c}\frac{x-2}{5}=\frac{y-0}{3}=\frac{z-\left(-3\right)}{-7}\\ \\ \frac{x-2}{5}=\frac{y}{3}=\frac{z+3}{-7}\end{array}$