Question

Sa se arate ca AX+BY+CZ=AY+BZ+CX (sunt vectori)

Answer 1

$\overrightarrow{AX}+\overrightarrow{BY}+\overrightarrow{CZ}=\overrightarrow{AY}+\overrightarrow{BZ}+\overrightarrow{CX} \\ \\ \overrightarrow{AX}-\overrightarrow{CX}+\overrightarrow{BY}-\overrightarrow{AY}+\overrightarrow{CZ} - \overrightarrow{BZ} = \overrightarrow{0}\\ \\ \Big(\overrightarrow{AX}+\overrightarrow{XC}\Big)+\Big(\overrightarrow{BY}+\overrightarrow{YA}\Big)+\Big(\overrightarrow{CZ}+\overrightarrow{ZB}\Big) =\overrightarrow{0}$
$\overrightarrow{AC}+\overrightarrow{BA}+\overrightarrow{CB} = \overrightarrow{0}$
$\overrightarrow{AC}+\Big(\overrightarrow{CB}+\overrightarrow{BA}\Big) = \overrightarrow{0}\\ \\ \overrightarrow{AC}+\overrightarrow{CA} = \overrightarrow{0}\\ \\ \overrightarrow{AC}-\overrightarrow{AC} = \overrightarrow{0}\\ \\ \overrightarrow{0}= \overrightarrow{0}\quad (A) \\ \\ \\ \Rightarrow \boxed{\overrightarrow{AX}+\overrightarrow{BY}+\overrightarrow{CZ}=\overrightarrow{AY}+\overrightarrow{BZ}+\overrightarrow{CX} }$