Analytic geometry: Fourth chapter space in the line and plane (2) space midpoint to the line, plane distance, the relationship between plane, straight and straight, straight and plane relations _ analytic geometry

The distance from the midpoint of a §4 space to a line or plane.

1. Distance from point to plane

The normal equation for 1o plane P is:

The distance from the point m (XO, Yo, zo) to the plane P is:

The general equation for the 2o plane P is:

The distance from the point m (XO, Yo, zo) to the plane P is:

2. Distance from point to line

The symmetric equation for the line L is:

(Straight line over M1 (x1, y1, Z1) point, direction number p, Q, R)

Then the distance of the point m (XO, Yo, Zo) to the straight line L is:

I,j,k is the unit vector on the three axes, the outermost "| |" A model that represents a vector.

or write as:

The interrelation between §5 plane

1. The intersection of the two plane

Plane p1:a1x+b1y+c1z+d1=0.

Plane p2:a2x+b2y+c2z+d2=0

Set φ to a two-face angle that intersects the planar P1 and the planar P2, then

2. Two plane parallel and vertical conditions

by 1, get

1o plane P1 and plane P2 parallel conditions:

2o plane P1 and plane P2 perpendicular to the condition:

3. Plane bundle

The pλ represents a planar bundle that passes through the known two plane P1 and the P2 intersection L. For a particular value lambda, the equation represents a plane in a planar bundle.

4. Three flat collinear conditions

Set three plane to

Then the condition of the three planar p1,p2,p3 collinear is the matrix

The rank is 2.

5. The plane

Pλμ represents a planar p1,p2,p3 of G through a three-plane intersection, and G is called the vertex of the pull. For any pair of fixed value λ,μ, the equation represents a plane in the plane.

6. Four plane common point conditions

Set four plane to

Then four plane P1, P2, P3, P4 the condition of the common point is determinant

§6 relationship between straight line and straight line and straight plane

1. Two-line angle

The symmetrical side of the two line is called

The direction number of straight line L1 and straight line L2 are P1, Q1, R1 and P2, Q2, R2,

The Yu Yingwei of the two straight line angle φ

2. Two straight line parallel and vertical conditions

By 1 can get two straight L1, L2 parallel and vertical conditions

3. The shortest distance between two not parallel lines

The shortest distance between the linear l1,l2 (equation and 1) is the distance between the common perpendicular of the l1,l2 and the intersection of the two lines.

One of the outermost "| |" Represents an absolute value.

4. Two linear coplanar conditions

By 3, the two linear l1,l2 coplanar condition is d=0, where the plane equation is

5. The angle between the line and the plane

Set Line

Plane

Then the sine of the line L and the plane p φ is

6. Parallel and vertical conditions of line and plane

By 5 The linear L and plane p are parallel and perpendicular to the condition

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