ࡱ> EGD~q` R bjbjqPqPCw::[%p8F|$%"!!!D%F%F%F%F%F%F%$&h')j%" !""j% %$$$":D%$"D%$$T$ D"$D%%0%$)~#)$)$!!$! !S!!!j%j%f$^!!!%""""  Angle between two vectors (1) Vector method In Diagram 1, let  EMBED Equation.3 ,  EMBED Equation.3  then  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  . (1) , where  EMBED Equation.3 . We see that when (i)  EMBED Equation.3  (ii)  EMBED Equation.3 .  (2) Co-geom. method As in the Diagram 2 on the right, the gradients of OA, OB are  EMBED Equation.3   EMBED Equation.3  Assuming  EMBED Equation.3 ,  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  , where  EMBED Equation.3 . . (2) Readers may investigate various cases for which the ( lies. (3) Check  Please check that (1) and (2) give the same result, that is, the following identity holds:  EMBED Equation.3    EMBED Equation.3   (4) Further investigations We use tan to get the formula in (2), can we use other trigo. functions? Refer to Diagram 2,  EMBED Equation.3   EMBED Equation.3  . (3) It is easy to see that (3) is good to obey the Pythagoras Theorem in Diagram 3. Readers may check:  EMBED Equation.3  gives (1). (5) Cosine Law If we apply cosine law for (OAB in Diagram 2, we have  EMBED Equation.3   EMBED Equation.3  . (4) Please check that (4) is equivalent to (1).     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