Question

Fie doua puncte fixe A,B si M un punct mobil in plan.Aratati ca expresia AB(vector)*AM(vector)+BA(vector)*BM(vector)nu depinde de pozitia punctului M.

Answer 1

$\overrightarrow{AB}\cdot \overrightarrow{AM} + \overrightarrow{BA}\cdot \overrightarrow{BM} = \\ \\ = \overrightarrow{AB}\cdot \overrightarrow{AM} +(-\overrightarrow{AB})\cdot \overrightarrow{BM} \\ = \overrightarrow{AB}\cdot \overrightarrow{AM} - \overrightarrow{AB}\cdot \overrightarrow{BM} \\ = \overrightarrow{AB}\cdot \Big(\overrightarrow{AM} - \overrightarrow{BM}\Big) \\ = \overrightarrow{AB} \cdot \Big(\overrightarrow{AM} - (-\overrightarrow{MB})\Big)$

$= \overrightarrow{AB} \cdot (\overrightarrow{AM}+\overrightarrow{MB})\\ = \overrightarrow{AB} \cdot \overrightarrow{AB} \\ \\ = 2\overrightarrow{AB}$

=> Expresia nu depinde de M.