. . Application of dot and cross products in plane Trigonometry We apply the concepts of dot and cross products of two vectors to derive a few formulae in plane trigonometry. Example .
(Cosine formulae) With usual notations, in any triangle ABC, prove the following by vector method. (i) bc (ii) ca B (iii) ab C With usual notations in triangle ABC , we have BC a CA and AB . Then | , | BC CA AB and BC CA AB = So, BC CA AB Then applying dot product, we get BC BC = ( ) ( CA AB CA AB ⋅− ⇒ BC CA AB CA AB ⇒ a = cos( bc ⇒ a = bc The results in (ii) and (iii) are proved in a similar way. Example .
With usual notations, in any triangle ABC , prove the following by vector method. (i) a C B (ii) b C (iii) c B Fig. . π –A π –C π –B C B Vector - - With usual notations in triangle ABC , we have BC a CA , and AB .
Then , | BC CA = , | AB and BC CA AB = So, BC CA AB Applying dot product, we get BC BC = − BC CA BC AB ⇒ | BC = − | | | cos( ) | | | | cos( BC CA C BC AB B ⇒ a = ab C ac B Therefore a C B . The results in (ii) and (iii) are proved in a similar way. Example . By vector method, prove that cos( β β β Let ˆ a OA and ˆ b OB be the unit vectors and which make angles α and β , respectively, with positive x -axis, where A and B are as in the Fig.
. . Draw AL and BM perpendicular to the x -axis. Then | | | | cos cos , | | | | sin OL OA LA OA α .