chapter 11 . math form 4 :(

Chapter 11 : Lines and planes in 3 - Dimensions

11.1 : Angles between Lines and Planes
* Identifying planes

• A plane is flat surface of an object.
• A two-dimensional shape has two dimensions which are length and breadth,and has only one plane.This shape has only area does not have volume.

          

• A three-dimensional shape has three dimensions which are length,breadth and height.It has more than one surface(planes or curved surfaces).This shape has both area and volume.

            




*Identifying horizontal,vertical and inclined planes


There are three types of planes :
(a) Horizontal plane - A plane that is parallel to the horizontal surface.

        
(b) Vertical plane - A plane that is perpendicular to the horizontal surface.

       
(c) Inclined plane - A plane that is inclined at an angle to the horizontal surface.
       


*Sketching three dimensional shapes

Three dimensional shapes can be drawn on grid papers or blank papers.The specific planes can then be identified as horizontal planes,vertical planes or inclined plane.

*Identifying lines that lie on or intersect with a plane

• In the diagram below,the line AB lies on the plane EFGH. Every point on the line AB lies to plane . 

            

• In diagram below, the line CD intersects the plane KLMN. The line CD meets the plane at only one point.

            


* Identifying normals to a plane 
A normal to a plane is a straight line which perpendicular to any line on the plane passing through the point of intersection of the line and the plane.

PQ is the normal to plane ABCD as shown below. 
            



*Orthogonal projections

The  orthogonal projection of a line PR on a plane,with point R on the plane,is the line joining R to the point of intersection of the normal from P to the plane,that is line RQ.
        


* The angle between a line and a plane

The angle between a line and a plane is the angle between the line and its orthogonal projection of the line on the plane.
     

* Solving problems

The solve problems involving the angles between a line and a plane,follow the steps below :

• Identify the normal to the given plane and the orthogonal projection of given line on the plane.
• Sketch the right-angled triangle involved.
• Identify the angle between the line and the plane.
• Solve the problem using Pythagoras' theorem and / or trigonometric ratios. 

11.2 : Angle Between Two Planes

*The line of intersection between two planes

Two planes, PQRS and RSUT meet at a straight line,RS, which is known as the line of intersection between the two planes.
          
*Drawing perpendicular lines to the line of intersection of two planes

To draw perpendicular lines to the line of intersection of two planes,follow the steps below.

• Draw the line of intersection of two planes.
• Mark a point on the line of intersection
• From the point, draw two lines, one on each plane which is perpendicular to the planes.

          

• RS is the line of intersection between the two planes.
• Line JK is on plane PQRS and perpendicular to RS.
• Line KL is on plane RSTU and perpendicular to RS.

*The angle between two planes

The angle between two intersecting planes in the angle between two lines,one each plane,drawn respectively from one common point on the line of intersection and is perpendicular to the line of intersection.

In the diagram below,QR the line of intersection of the planes, PQR  and QRST. PM and MN are perpendicular to the line QR at M.