Question

Vectors

Prove that (a x b) .
(c + xa + yb) = a.(b x c) for any numbers x and y.

Answer 1

prove\[\forall x,y\in\mathbb R:(\vec a\times\vec b)\cdot(\vec c+x\vec a+y\vec b)=\vec a\cdot(\vec b\times \vec c)\]is this the problem?

Answer 2

Yes. Sorry, I struggled with the notation.

Answer 3

im thinking that \(\vec a \times \vec b\) is perpendicular to \(\vec a\) and \(\vec b\)

Answer 4

good call :D

Answer 5

:)

Answer 6

....forgive me, you may need to explain this to me...

Answer 7

note that after expanding we have\[(\vec a\times\vec b)\cdot(\vec c+x\vec a+y\vec b)=(\vec a\times\vec b). \vec c+x(\vec a\times\vec b).\vec a+y(\vec a\times\vec b).\vec b\]

Answer 8

...and what is the product of\[(\vec a\times\vec b)\cdot\vec a\]?

Answer 9

0?

Answer 10

Ah. So you're left with the triple scalar product abc. Thanks