Plane and its equation

We call any straight line perpendicular to a plane the normal of this plane.

Let a given fixed point be Po(x0, y0, z0) and a set of directions of a given normal n be {A, b, C}A2+B2+C2≠0, then the plane equation passing through this fixed point and taking n as the normal can be expressed as:

Note: This form of equation is called the point method of plane equation.

Example: Let the direction number of the straight line L be {3, -4, 8}, and find the plane equation that passes through the point (2, 1, -4) and is perpendicular to the straight line L. 。

Solution: The plane equation obtained by applying the above formula is:

that is

We arrange the form as follows:

Ax+By+Cz+D=0。

It's called the general formula of plane equation. Where the coefficients a, b and c of x, y, z, y and z are a set of direction numbers of the plane normal.

Equations of several special position planes

1, through the origin

The general form of its plane equation is:

Ax+By+Cz=0。

2, parallel to the coordinate axis

The general form of the plane equation parallel to the X axis is:

By+Cz+D=0。

The general form of the plane equation parallel to the Y axis is:

Ax+Cz+D=0。

The general form of the plane equation parallel to the Z axis is:

Ax+By+D=0。

3, through the coordinate axis

The general form of the plane equation passing through the X axis is:

By+Cz=0。

The general form of the plane equation passing through the Y axis and the Z axis is:

Ax+Cz=0，Ax+By=0。

4, perpendicular to the coordinate axis

The general form of the plane equation perpendicular to the X, Y and Z axes is:

Ax+D=0，By+D=0，Cz+D=0。

Straight line and its equation

Any given straight line has a definite direction. However, there can be an infinite number of straight lines with definite directions, which are parallel to each other. If a straight line is required to pass through a certain point, the straight line will be uniquely determined, so the equation of the straight line can be expressed by the number of directions passing through it and the coordinates of the fixed point.

Let the direction number of the straight line L be {l, m, n} and a point Po(x0, y0, z0) on L, then the equation of the straight line L can be expressed as:

The above formula is the equation of straight line L, and the form of this equation is called the symmetric formula of straight line equation.

The linear equation also has a general formula, which is obtained by two plane equations at the same time, as follows:

This is the general formula of linear equation.

Parallel and vertical relationship between plane and straight line

For a given plane, its normal can also be known. Therefore, the parallel and vertical relationship between planes is transformed into the parallel and vertical relationship between straight lines. The parallel and vertical relationship between plane and straight line, that is, the parallel and vertical relationship between plane normal and straight line.

Generally speaking, the vertical parallel relationship between a plane and a straight line is finally transformed into the parallel vertical relationship between straight lines. I won't list examples here.