§4 Lines and planes in space

§4 Lines and planes in space

The direction of the straight line

name and description	graphics
[ direction angle ] The angle a , b , g of the straight line OM passing through the origin O and the three coordinate axes is called the direction angle of the line ( the direction of OM is the direction away from the origin O ) : a = ∠ MOx , b = ∠ MOy , g = ∠ MOz [ direction cosine ] The cosine of the direction angle of a line is called the direction cosine: , , In the formula , l ² + m ² + n ² = 1
[ Number of directions ] The coordinates ( p , q , r ) of any point W on the straight line OM passing through the origin and parallel to the straight line L are called the number of directions of the straight line L, and is the direction cosine of the straight line OM
name and description	graphics
[ Direction cosine of a line passing through two points ] , , in the formula At this time, the positive direction of the straight line is the direction from M ₁ ( x ₁ , y ₁ , z ₁ ) to M ₂ ( x ₂ , y ₂ , z ₂ ) .

The equation of the plane

Equations and Graphics		Description
[ intercept ]		a, b, c are called the intercepts of the plane on the three coordinate axes, respectively
[ dot French ] ( A , B , C are not equal to zero at the same time )		The plane passes through the point M ( x ₀ , y ₀ , z ₀ ) and the number of directions of the normal N is A , B , C
[ Three-point type ]		The plane passes through three points : M ₁ ( x ₁ , y ₁ , z ₁ ) M ₂ ( x ₂ , y ₂ , z ₂ ) M ₃ ( x ₃ , y ₃ , z ₃ )
or =0
equation	with graphics	Description
[ General formula ] Ax + By + Cz + D = 0 ( A , B , C are the direction numbers of the normal of the plane, and are not equal to zero at the same time )		When D = 0 , the plane passes through the origin When A = 0 ( or B = 0 , or C = 0) , the plane is parallel to the x - axis ( or y - axis, or z - axis ) When A = B =0 ( or A = C =0 , or B = C =0) , the plane is parallel to the Oxy plane ( or Ozx , or Oyz )
[ normal type ] ( a , b , g are the direction angles of the normal line of the plane, p 3 0 is the length of the normal line, that is, the distance from the origin to the plane )		The general formula of the plane can be transformed into the normal formula , which is called the normalization factor of the plane. When D < 0 , the positive sign is taken; when D > 0 , the negative sign is taken.
[ vector ] ( r - r₀ ) × a = 0		The plane passes through the end point of the vector radius r₀ and is perpendicular to the known vector a , r is the vector radius of any point on the plane

The equation of a straight line

Equations and Graphics

Description

[ General formula ( or face-to-face )]

Taking the straight line L as the intersection of two planes, its number of directions is

[ Symmetrical ( or parametric )]

The straight line L passes through the point M ( x ₀ , y ₀ , z ₀ ) and has the number of directions p , q , r

Equations and Graphics

Description

[ two-point type ]

The straight line L passes through two points M ₁ ( x ₁ , y _{1 ,}z ₁ ) and M ₂ ( x ₂ , y ₂ , z ₂ )

[ projective ]

The straight line L is the intersection of the two planes y = ax + g and z = bx + h ; it passes through the point (0, g , h ) and has direction numbers 1, a , b

[ vector ]

r = r₀ + t a

(- ￥ < t < ￥ )

The straight line L passes through the end point of the vector radius r₀ and is parallel to the known vector a , where r is the vector radius of any point on L

4. Interrelationship between points, lines and planes in space

Equations and Graphics

Formula and Explanation

[ Included angle between two planes ]

P ₁A ₁x + B ₁y + C ₁z + D ₁ = 0

P ₂ A ₂x + B ₂y + C ₂z + D ₂ = 0

where is the dihedral angle of the two planes P ₁ and P ₂

Equations and Graphics

Formula and Explanation

[ Condition of plane bundle × collinearity of three planes ]

P _l ( A ₁x + B ₁y + C ₁z + D ₁ ) + l ( A ₂x + B ₂y + C ₂z + D ₂ ) = 0

( l is a parameter, - ￥ < l < ￥ )

[ Condition of plane handle × four planes co-point ]

P _l_m ( A ₁x + B ₁y + C ₁z + D ₁ ) +

l ( A ₂x + B ₂y + C ₂z + D ₂ ) +

m ( A ₃x + B ₃y + C ₃z + D ₃ )

= 0

( l , m are two independent parameters,

- ￥ < l , m < ￥ )

For a definite value of l, P l _represents an intersection of two planes P ₁ and P ₂

The plane of the line L , when l takes all values, the whole of the plane represented by P _{l passing through}L is called the plane beam, and L is called the axis of the beam .

Let P ₃ be A ₃x + B ₃y + C ₃z + D ₃ = 0 , then the condition for the collinearity of the three planes P ₁ , P ₂ , P _{3 is a matrix}

The rank of is equal to 2.

For a pair of definite values of l , m , P _l_m represents a plane passing through the intersection G of the three planes P ₁ , P ₂ and P ₃ , when l , m take all values, P _l_m represents the plane passing through G The whole of is called the plane handle, and G is called the vertex of the handle .

Assuming that P ₄ is A ₄x + B ₄y + C ₄z + D ₄ = 0 , the condition for the common points of the four planes P ₁ , P ₂ , P ₃ , and P ₄ is the determinant

[ distance between points and faces ]

normal

x cos a + y cos b + z cos g - p = 0

General formula Ax + By + Cz + D = 0

d _method = | x ₀ cos a + y ₀ cos b + z ₀ cos g - p |

where d is the distance from point M ( x ₀ , y ₀ , z ₀ ) to the plane

Equations and Graphics

Formula and Explanation

[ distance of dotted line ]

where d is the distance from the point M ( x ₀ , y ₀ , z ₀ ) to the straight line L , i , j , k are the unit vectors on the three coordinate axes, and the outermost symbol “ | | ” represents the modulus of the vector

[ Included angle between two straight lines ]

L ₁

L ₂

where j is the angle between the two straight lines L ₁ and L ₂

[ The shortest distance between two non-parallel lines ]

L ₁

L ₂

The so - called shortest distance refers to the distance between the common vertical line of L ₁ , L ₂ and the intersection of the two lines . The equation of the plane is