Symmetric about the origin: (-x,-y,-z)

Symmetry about the coordinate axis: symmetric about whom it is unchanged, such as symmetry about the x-axis (x,-y,-z)

Symmetric about the Qxy plane, (x,y,-z) Symmetric about the Oyz plane, (-x,y,z) Symmetric about the Oxz plane. (x,-y,z)

Position vector of point: Take a point O as the base point, any point P is represented by OP, OP is the position vector of point P

The vector expression of a straight line in space: straight line l, points A and B are on it, and any point O and point P are on the straight line l. The necessary and sufficient condition is that there is a real number t, so that OP=OA tAB

Vector expression of the space plane: space plane ABC, any point O, point P is located on the plane ABC. The necessary and sufficient condition is that there are real numbers x, y, so that OP=OA xAB yAC

Lines are parallel

Lines and planes are parallel

Face to face parallel

vertical line

vertical line

Face to face vertical

The distance from the point to the straight line: PQ=√（∣AP∣²-∣AQ∣²)=√[a²-(a∙u)²], u represents the unit direction vector of the straight line l

The distance from the point to the plane: PQ=∣AP∙n∣/∣n∣, the equal volume method and the equal area method can also be used. When the straight line is in the plane or intersects the plane, d=0. In the case of parallelism, both line-surface distance and surface-surface distance can be converted into point-surface distance.

Angle ɑ formed by straight lines of different planes: If the direction vectors of straight lines l₁ and l₂ are u₁ and u₂ respectively, then cosɑ=∣cos<u₁,u₂>∣

The angle ɑ formed by the straight line and the plane: If the direction vector of the straight line l is u and the normal vector of the plane is n, then sinɑ=∣cos<u,n>∣

The angle between the plane and the plane: If the normal vectors of the two planes are n₁ and n₂ respectively, then cosɑ=∣cos<n₁,n₂>∣, ɑ≤90⁰