If "$\vec a\times\vec b=\vec c \times \vec d$" . And $\vec a\times \vec c = \vec b \times \vec d$ Then prove that $\vec a + \vec d = k(\vec b +\vec c)$, where $k$ is some number.
I got as far as proving $\vec a - \vec d = k'(\vec b - \vec c) $
But no further progress..
Note that $a$, $b$, $c$, $d$ are vectors and $k$ is any real no.
Hint:$$(\vec a+\vec d)\times(\vec b+\vec c)=\vec a\times\vec b+\vec a\times\vec c+\vec d\times\vec b+\vec d\times\vec c$$