THE POLAR CO-ORDINATES OF A POINT 75 



These relations give the direction cosines of the perpendicular 

 to this plane. 



If is the angle between the two planes, then this will also be 

 tin angle between the two perpendiculars, 



and cos = / 1 / a + w^wij + njn z 



! P&t t PiPz 



( I J_ 1 i 



Ia x a 2 fcj&jj CjC 2 J 



Example. Find the angle between the planes 21a? + 35y + 15 

 105 and I5x - 9y + 5z - 45. 



Then for the first plane - + ^ + - = 1 

 537 



2 1 



and 



_ _ 



25 9 49 

 p l = 2-414 



For the second plane - - + - = 1 



and 



9 25 81 

 p 2 = 2-474 



Then cos 6 = 2-414 x 2-474 j^--^ + 



= 0-0948 

 8 = 84 34' 



48. The Polar Co-ordinates of a Point. 

 The polar co-ordinates of a point P are : 



r, the distance the point is from the origin. 



6, the angle between the plane containing OP and the 

 plane ZOX. 



<j>, the angle OP makes with the axis OZ. 

 Then CPO is a triangle, right angled at C 

 Hence OC = r cos <f>, and PC = r sin <f> 



But OQ = PC = r sin < 



Also AOQ is a triangle, right angled at A 



AO 



Then -^ - = cos 



\j\^ 



and AO = r cos sin <f> 



