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

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,

$\vec{a}=3 \hat{i}-2 \hat{j}-5 \hat{k}$

The direction ratios of PQ are given by,

$(3-3)=0,(-2+2)=0,(6+5)=11$

$\vec{b}=0 . \hat{i}-0 . \hat{j}+11 \hat{k}=11 \hat{k}$

The equation of $\mathrm{PQ}$ in vector form is given by, $\vec{r}=\vec{a}+\lambda \vec{b}, \lambda \in R$

$\Rightarrow \vec{r}=(3 \hat{i}-2 \hat{j}-5 \hat{k})+11 \lambda \hat{k}$

The equation of PQ in Cartesian form is

$\frac{x-x_{1}}{a}=\frac{y-y_{1}}{b}=\frac{z-z_{1}}{c}$ i.e.,

$\frac{x-3}{0}=\frac{y+2}{0}=\frac{z+5}{11}$