3. 7 relationship between the position of a straight line and a plane

7 relationship between the position of a straight line and a plane

 

IResolution conditions for positional relationships:

Set line L: With plane π: Ax + By + Cz + D = 0

Then, the intersection of L and π is unique

A (+ X) + B (+ Y) + C (+ Z) = 0 <clerk> AX + BY + CZ =0

The substring has the intersection of L and π <clerk> AX + BY + CZ =0;

L ∥ π <strong> there is no unique t to make (+ tX) + B (+ tY) + C (+ tZ) + D = 0

<Strong> AX + BY + CZ = 0

L on π, there are an infinite number of t numbers in the "struct" to make A (+ tX) + B (+ tY) + C (+ tZ) + D = 0

<Strong> AX + BY + CZ = A + B + C + D = 0

Inference: L ∥ π, but L is not on π <clerk> AX + BY + CZ = 0, but A + B + C =0

 

IIIntersection of line and plane:

In the Cartesian system, the equations of the linear L and the plane π are as follows: V {X, Y, Z} and n {A, B, c} is the direction vector of l and the normal vector of π,

(Fig. 3.6)

Let round (L, π) =, round (v, n) = θ then

= θ OR = θ-(θ is an acute angle)

∴ Sin = ± cos θ = ∣ cos θ cosine =

Example: Returns the equation of a straight line having a point (-1, 2,-3) parallel to a plane 6x-2y-3z + 1 = 0 and intersecting with a straight line.

Solution: Omitted.