The equation that can represent all straight lines u(x-x0)+v(y-y0)=0(u, v are not all zero) is one of the linear equations in high school mathematics. (x-x0) u = (y-y0) v, u and v are not all zero, which is called the point normal equation and can represent all straight lines.

The coordinates of any point on the plane π satisfy this equation. The points whose coordinates satisfy the equation are all on π, so this equation is the equation that plane π passes through this point and is perpendicular to the vector, which is called the point normal equation of the plane.

Characteristics of point-method equation

A plane π can be completely determined by any point on π and any vector perpendicular to π. Any vector perpendicular to π is called the normal vector of π.

The formula of point normal is determined by the coordinates of a point on a straight line. The normal vector of (x0, y0) and this straight line is a point on the straight line, and {u, v} is the normal vector of the straight line.

For example, finding an equation of a straight line passing through point A (2, 3) applies to the following conditions respectively:

(1) is parallel to the straight line 2x +y -5=0.

(2) perpendicular to the straight line x -y -2=0[3][2]

Solution: (1) Because the normal vector of the straight line 2x +y -5=0 is a =(2, 1), and because the straight line and the straight line are found,

2x +y -5=0 is parallel, so a is also the normal vector of a straight line, which is obtained from the French equation of the straight line point:

2(x -2) + 1？ (y -3) =0

So the linear equation is 2x +y -7=0.

(2) Since the normal vector of the straight line x -y -2=0 is m =( 1,-1), let the normal vector of the straight line be n =(x 0, y 0).

The topic is m ⊥n, so m? N =0 means X 0-Y 0 = 0, that is, X 0 = Y 0, so the normal vector of the straight line is N = (X 0, X 0), which is obtained from the French equation of the straight line point: X 0 (X-2)+X 0 (Y-3) = 0 (X.

So the linear equation is: x +y -5=0.

Note: When applying the normal equation of straight line points, the key to solve this kind of problem is to determine the point and normal vector.