3.3. Obtain the equation of the line passing through two given points $P_{0}(1 ; 2)$ and $P_{1}(3 ; 2)$.
Solution.

1st step. Make a schematic drawing (Fig. 3.3).

2nd step. Write down the coordinates of the vectors defining the given geometric object - the line. This is the current vector lying on the line $\overrightarrow{P_{0} M}=\{x-1 ; y-2\}$ and the vector formed by two points on the line

$$
\overrightarrow{P_{0} P_{1}}=\{3-1 ; 2-2\}=\{1 ; 0\} .
$$

3rd step. Write down one of the conditions - orthogonality or collinearity, according to the problem. According to the problem, $\vec{P}_{0} M \| \bar{P}_{0} \vec{P}_{1}$, so we will use the condition of collinearity of two vectors in coordinate form $(x-1) / 2=(y-2) / 0$. We obtain the canonical equation of the line. Sometimes in the answer to the considered problem, it is mistakenly not written that part of the equation where zero appears in the denominator, referring to the fact that division by zero is not allowed. However, the equation $(x-1) / 2=(y-2) / 0$ does not describe division by zero; here only a symbolic notation is made that the current vector $\overrightarrow{P_{0} M}$ is collinear with the vector $\widetilde{P}_{0} \vec{P}_{1}$, one of whose components is zero.

![](https://cdn.mathpix.com/cropped/2024_05_22_f5a9bb7960904462604bg-113.jpg?height=179&width=459&top_left_y=1531&top_left_x=146)

Fig. 3.2

![](https://cdn.mathpix.com/cropped/2024_05_22_f5a9bb7960904462604bg-113.jpg?height=162&width=448&top_left_y=1531&top_left_x=671)

Fig. 3.3