The Angle between two Planes; an application of the Angle between two Lines

Problem:

Find the angle between the plane given by the vector equation r1.[3, 4, 0]  = 5 and the plane given by r2.[1, 2, 3]  = 6
A diagram of this is shown on the right. [Graphics:Images/index_gr_1.gif] is a normal vector to Plane 1

[Graphics:Images/index_gr_2.gif] is a normal vector to Plane 2.

θ is the angle between the two planes.

By simple geometrical reasoning;
    angle between two planes equals angle between their normals.
The angle, θ, between the two normal vectors can be easily found using  'the angle between two lines'  method.



From the equations to the two given planes,[Graphics:Images/index_gr_3.gif].[3, 4, 0]  = 5 and,  [Graphics:Images/index_gr_4.gif].[1, 2, 3]  = 6, the normal to Plane 1 is parallel to the vector [3, 4, 0] and the normal to Plane 2 is parallel to the vector [1, 2, 3].


The angle between the two normals is therefore, the angle between the two vectors [3, 4, 0] and [1, 2, 3];

                             [3, 4, 0] . [1, 2, 3] = | [3, 4, 0] | × | [Graphics:Images/index_gr_5.gif] |cosα


                                                     cosα =  [Graphics:Images/index_gr_6.gif]    
                                                       
                                                            α = [Graphics:Images/index_gr_7.gif][Graphics:Images/index_gr_8.gif]
                                                            
                                                                =  0.94 radian
                                                                
                                                                
The required result, the angle between the two planes is therefore 0.94 radian;
 

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