§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


[ Threepoint 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  r_{ 0} ) × a = 0


The plane passes through the end point of the vector radius r_{ 0} 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 facetoface )] L _{}

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

[ Symmetrical ( or parametric )] _{} or 
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 

[ twopoint 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 ] L _{} 

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 = r_{ 0} + t a ( ￥ < t < ￥ )


The straight line L passes through the end point of the vector radius r_{ 0} 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 _{1 }A _{1 }x + B _{1 }y + C _{1 }z + D _{1} = 0 _{}_{}_{}_{} P _{2 } A _{2 }x + B _{2 }y + C _{2 }z + 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 _{1 }x + B _{1 }y + C _{1 }z + D _{1} ) + l ( A _{2 }x + B _{2 }y + C _{2 }z + D _{2} ) = 0 ( l is a parameter,  ￥ < l < ￥ )
[ Condition of plane handle × four planes copoint ] P _{l }_{m} ( A _{1 }x + B _{1 }y + C _{1 }z + D _{1} ) + l ( A _{2 }x + B _{2 }y + C _{2 }z + D _{2} ) + m ( A _{3 }x + B _{3 }y + C _{3 }z + 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 _{3 }x + B _{3 }y + C _{3 }z + 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 _{4 }x + B _{4 }y + C _{4 }z + 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 ] L _{}

_{} 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 nonparallel 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 _{}



Equations and Graphics 
Formula and Explanation 
[ The angle between the line and the plane ] L _{} P Ax + By + Cz + D = 0

_{} where j is the angle between the straight line L and the plane P

[ Parallel and perpendicular conditions of straight lines and planes ]
parallel condition 
vertical condition 
line to line _{} face to face _{} line and surface _{}

p _{1 }p _{2} + q _{1 }q _{2 } + r _{1 }r _{2} = 0
A _{1 }A _{2 } + B _{1 }B _{2} + C _{1 }C _{2 } = 0
_{}
