3.1. Obtain the equation of a straight line in the plane, given a point $P(2 ; 3)$ through which the line passes, and a vector perpendicular to the line $\vec{n}=\{A ; B\}=\{4 ;-1\}$.
Solution.

1st step. Make a schematic drawing (Fig. 3.1).

2nd step. Write down the coordinates of the vectors defining the given geometric object - the line. This is the normal vector of the line

$$
\vec{n}=\{A ; B\}=\{4 ;-1\}
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_f5a9bb7960904462604bg-112.jpg?height=360&width=454&top_left_y=1067&top_left_x=670)

Fig. 3.1

and the current vector $\overrightarrow{P M}=\{x-2 ; y-3\}$.

3rd step. Write down one of the conditions - orthogonality or collinearity, according to the problem. According to the problem, $\overline{P M} \perp \vec{n}$, so we will use the condition of orthogonality of two vectors in coordinate form

$$
4(x-2)+(-1)(y-3)=0
$$

4th step. Perform algebraic transformations to reduce the equation to a known form. We obtain the general equation of the line $4 x-y-5=0$.