Question

Find the vector and the Cartesian equations of the line that passes through the points $(3,-2,-5),(3,-2,6)$

Answer 1

Let the line passing through the points $P(3,-2,-5)$ and $Q(3,-2,6)$ be $PQ$. Since $PQ$ passes through $P(3,-2,-5)$, its position vector is given by,
$\overrightarrow{a}=3\hat{i}-2\hat{j}-5\hat{k}$
The direction ratios of $PQ$ are given by,
$(3-3)=0,(-2+2)=0,(6+5)=11$
The equation of the vector in the direction of $PQ$ is
$\overrightarrow{b}=0\hat{i}-0\hat{j}+11\hat{k}=11\hat{k}$
The equation of $PQ$ in vector form is given by $\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b},\lambda \in R$
$\Rightarrow \overrightarrow{r}=(3\hat{i}-2\hat{j}-5\hat{k})+11\lambda \hat{k}$
The equation of $PQ$ in Cartesian form is
$\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}$
$\Rightarrow \frac{x-3}{0}=\frac{y+2}{0}=\frac{z+5}{11}$