Three-Dimensional (3D) Space
     Points, Lines and Planes in Three-Dimensional (3D) Space
      Position of two lines in a three-dimensional space
         Coplanar lines and skew lines
      Line and plane in a three-dimensional space
      Two planes in a three-dimensional space
         The line of intersection of two planes
      Orthogonality of line and plane
      Orthogonality of two planes
      Orthogonal projection of a point onto a plane
         The distance from a point to a plane
      Angle between a line and a plane, and angle between two planes
Position of two lines in a three-dimensional space
Exactly one line passes through two different points in a space. 
Three points not on the same line determine a plane.
A plane is also determined by:
    -a line and a point not on the line;      -a point and a line, which is normal/perpendicular to the plane; 
    -two lines which intersect;                 -two lines which are parallel. 
Two lines are parallel if they do not intersect and they are coplanar (lie on the same plane). 
Two lines which do not intersect but which are not coplanar are called skew lines.
Skew lines do not lie in a single plane together.
 
The relations between points, lines and planes are represented by sides, edges, diagonals and vertices of the
rectangular parallelepiped (cuboid) shown in the pictures below.
   
Line and plane in a three-dimensional space
A line l intersects a plane P at a point A, as shows the left picture.
A line is defined to be parallel to a plane if the line and the plane are disjoint (empty intersection). That is,
there is a line in the plane which is parallel to the given line, as shows the right picture.
The line k below lies in the plane P1 and P2 .
 
Two planes in a three-dimensional space
Two distinct planes either are disjoint (empty intersection) or intersect in a line.
Two disjoint planes are called parallel.
   
Orthogonality of line and plane
A line is perpendicular to a plane if it is perpendicular to every line in the plane that passes through its
intersection point.
The plane P2 is perpendicular to the plane P1 if there is a line in P2 which is perpendicular to the plane P1 .
   
Orthogonal projection of a point onto a plane
Orthogonal projection of a point P onto a plane is the intersection point P' of the perpendicular drawn from P
to the plane of projection.
Orthogonal projection of a line segment is a line segment or point depending of the position of the line
segment in relation to the plane of projection.
The projection will be a point if the line segment is perpendicular to the plane of projection.
 
The distance from a point to a plane is the distance from the point to its orthogonal projection to the plane,
i.e.,  AA'  = d(A,A').
For example, if the surface of a triangle and the plane of projection are mutually perpendicular, as is shown in
the picture below, then the projection of the triangle is a line segment, otherwise the projection is a triangle.
 
Angle between a line and a plane, and angle between two planes
The angle between a line and a plane is the angle between the line and its orthogonal projection to the plane. 
The angle between two planes is the angle between two lines, one lying in each plane, drawn perpendicular to
the intersection of the planes at the same point, as is shown below.
 
Beginning Algebra Contents
Copyright © 2004 - 2012, Nabla Ltd.  All rights reserved.