Analytic geometry: Fourth chapter space Line and Plane (1) Space line direction, plane equation, space linear equation _ analytic geometry

§1 the direction of the space line

1. Direction angle

The angle between the straight OM and the three axis through the Origin o α,β,γ is called the direction angle of the line

namely: Α=∠mox,β=∠moy,γ=∠moz

2. Direction cosine

The cosine of the direction angle of the line is called the direction cosine.

L=cosα=x/ρ, m=cosβ=y/ρ, n=cosγ=z/ρ

which

，

3. Direction number

If L is any line in space, and a straight line om,om from the Origin o to a straight line L, the coordinate of the point W is (P,q,r), then the number of directions of the P,q,r called the straight line L.

In fact, the direction of the straight L and straight om Yu Yingwei

l=p/k, m=q/k, n=r/k

which

4. The direction cosine of a straight line over two points

Set straight L over point M1 (x1, Y1, Z1) and M2 (x2, Y2, Z2), then line L Direction Yu Yingwei

l=cosα= (X2-X1)/D, m=cosβ= (y2-y1)/D, n=cosγ= (Z2-Z1)/d

which

　

§2 plane equation

1. Intercept type

The a,b,c in the formula is the intercept of the plane on the three axes respectively.

2. Dot French

(0 a,b,c when different)

The plane passes the point m (x0,y0,z0), and the direction number of the normal n is a,b,c.

　

　

3. Three-point

Or

The plane passes through three points M1 (x1, Y1, Z1), M2 (x2, Y2, Z2), M3 (X3, Y3, Z3).

　

　

　

4. General type

(0 a,b,c when different)

The normal direction number of the plane is a,b,c.

Special case:

⑴ when D=0, the plane passes the origin point.

⑵ when a=0 (or b=0, or c=0), the plane is parallel to the x-axis (or y-axis, or z-axis).

⑶ when a=b=0 (or a=c=0, or b=c=0), the plane is parallel to the Oxy plane (or OZX plane, or the Oyz plane).

The ⑷x=0 (or y=0, or z=0) represents the Oyz plane (or oxz plane, or oxy plane).

5. Normal type

Where the α,β,γ is the direction angle of the plane normal, the p≥0 is the normal length, i.e. the distance from the origin point O to the plane.

Note. The general equation of the plane ax+by+cz+d=0 can be converted to normal type:

In-style

A d<0 called a planar factor that takes a positive sign when it is d>0, and a minus when the time is taken.

6. Vector type

(r-r0) α=0

In the formula, R is the vector path of any point on the plane. The plane is perpendicular to the known vector α by r0 the end of the known sagittal path.

Linear equation of §3 space

1. General (or cross-face)

L

The line L is the intersection of two planes, its direction number is

，，

2. Symmetrical type (or parameter type)

L

or L: (-∞

The line passes through the point m (x0, y0, z0) and has a number of orientations p, Q, R.

　

3. Two-point

L

Straight line L through Point M1 (x1, Y1, Z1) and M2 (x2, Y2, Z2) two points.

　

　

4. Projective type

L

The straight line L is the intersection of planar y=ax+g (parallel to the z axis) and planar z=bx+h (parallel to the Y axis); L through the dots (0, G, h), with the direction number 1, a, B.

5. Vector type

L:r=r0+tα (-∞

The line L passes through the r0 of the vector path and is parallel to the known vectors α, and r is the vector path of any point on L.

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