Question

assuming U, V and W are vectors, prove that
(UxV)*[(VxW)x(WxU)]=[U*(VxW)]^2,
where
x is cross product,
* is dot product.

Answer 1

i know Ax(BxC)=B(A.C)-C(A.B), and A.(BxC)=B.(CxA)=C.(AxB),
but i dont know where to start from.

Answer 2

im not seeing the v u ws right, they tend to merge into one .....

Answer 3

ugh o.o?

Answer 4

i assume we have to do it with identities then?

Answer 5

the question hinted to use Ax(BxC)=B(A*C)-C(A*B), and A*(BxC)=B*(CxA)=C*(AxB), but i cant see where's the catch

Answer 6

is the dot product distributive?

Answer 7

a.(bxc) = a.b ? a.c

Answer 8

dot produce a scalar, and crossing a scalar is undefined so i really got to read up on these properties again :)

Answer 9

er yes dot is distributive and commutative, scalar is not commutative and associative

Answer 10

scalar is not commutative and not* associative

Answer 11

erm sry not scalar, its cross, lol

Answer 12

\[\vec a.(\vec b\times \vec c)=\vec a\vec c-\veca \vec b \]

oy this thing is a headache :)

Answer 13

er izit possible to let UxV be A, VxW be B, and WxU be C,

Answer 14

(UxV) * ( (VxW)x(WxU) ) = ( U*(VxW) )^2

(WxU)*((UxV)x (VxW)) - (VxW) ( (UxV)x(WxU) ) = ( U*(VxW) )^2

yes, its prolly best to redifine them so that he mess cleans up; since all they produce are another vector

Answer 15

(UxV) * ( (VxW)x(WxU) ) = ( U*(VxW) )^2

A.(BxC) = (U.B)^2

the thing is now we have to define "U" from the others in order to pair it up in the end

Answer 16

but once you get thist o something close in resemblance; you can switch back to the UVW mess and see if it pans out

Answer 17

erm does (VxW)x(WxU) = VxWxWxU = VxWxU?

Answer 18

kinda mixed up with boolean already LOL

Answer 19

A.(BxC) = C.(AxB) - B.(AxC)

WxW is only 1, if W = W^-1

Answer 20

in general, WxW not= W perse

Answer 21

and since there is no indication that these are perp to each other we should keep it as general as possible

Answer 22

just my thoughts about it tho

Answer 23

ouh i'll try again.

Answer 24

i cant parse it with any degree of certainty :) so i wish you the best of luck on it; and when all else fails, google it :)

Answer 25

cant google it, lol. maximum only managed to obtain the 2 properties as given in the question。

Answer 26

anyway thx for your opinion ;D