1. Chapter 2 Planes and Lines
10. Equation of a plane
It contains three points not on a line
It contains every point on any line joining two points on it
It does not contain all the points of a space
Theorem. The locus of the points whose coordinates satisfy a linear equation
Ax+By+Cz+D=0
With real coefficients is a plane

2. 11. Plane through three points

3. The reason of the above equation:
Consider the linear (uniform order) equation with respect to variables A, B, C and D, these four variables are none zero, thus the coefficients of the linear equation must be zero:
Ax+By+Cz+D=0

4. 12. Intercept form of the equation of a planeZ c O X Y

5. Exercises P14, 1and 4
13. The normal form of a plane

6. 14. Reduction of the equation of a plane to the normal form:
By normalizing vector (A,B,C), we have

7. Theorem Two points P and Q are on the same side or opposite sides of plane Ax+By+Cz+D=0,
according as their coordinates make the first member of the equation of the plane have the same or opposite signs.

8. 15. Angle between two planes
Theorem. The cosine of the angle between two planes
is defined by the equation:
In particular: perpendicular and parallel if:

9. Distant to a point from a plane
Normalize the equation and substitute the coordinates of point in the plane:

10. Direction cosines of the intersection line of two planes
Let two planes be:

11. Forms of the equations of a line
Suppose that the line pass through a point P1 and has direction cosines V , then
P-P1 = tV or

12. Parametric equation of a line
Here V is a vector and is a fixed point

13. Fraction form of a line

14. Angle which a line makes with a plane
Let the line and plane are respectively as follows:

15. Then N V

16. Distance from a point to a line.
Exercises P23, 1, 6,12,13.
Distance from a point to a line.
Given the line
and the point
not lying on it. Find the distance between the point and the line.

17. As in the following figure, we need only to compute
Therefore the result follows P

18. 23. Distance between two non-intersecting lines
(notice that distance is zero if they intersect)
Let the two lines be given as follows:

19. G H
We must find a line GH which is perpendicular to both of the given lines

20. 24. System of planes through a line
Then the planes passes through the line , is called the parameter of the pencil of planes

21. Homogeneous coordinates of point and plane
convenient to express the coordinates of a point in terms of four numbers, x’,y’,z’,t’ by means of the equation：
A set of four numbers (x’,y’,z’,t’), not all of which are zero, that satisfy the above equation are called homogeneous coordinates of a point

22. Exercises: P28, No. 1 and 3