SOLUTION: If addition of three vectors U, V and W is zero i.e. if U + V + W = 0 then UxV = VxW = WxU where x means cross product.

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Question 1188263: If addition of three vectors U, V and W is zero i.e. if U + V + W = 0 then UxV = VxW = WxU where x means cross product.
Found 2 solutions by Alan3354, ikleyn:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
If addition of three vectors U, V and W is zero i.e. if U + V + W = 0 then UxV = VxW = WxU where x means cross product.
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You didn't ask a question.
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If the sum of any 3 vectors is zero, they have to be colinear.
---> the cross products of any pair = zero.
Answer by ikleyn(52776)   (Show Source): You can put this solution on YOUR website!
.

We consider these vectors as 3D vectors in 3D vector space.


Since the sum of the three vectors U, V and W,  U + V + W is zero,  U + V + W = 0,

it means that the three vectors U, V and W are  coplanar,  i.e. lie in one plane in 3D space.


All cross products U x V,  V x W  and  W x U are vectors perpendicular to this plane.


The magnitude (= the modulus) of the cross product to each pair of these vectors is the area of a parallelogram, 
spanned by the corresponding pair of vectors.


Next, these three pairs of vectors actually span THE SAME parallelogram.

THEREFORE, the magnitude (the modulus) of each cross product is the same for all three cross products.


Finally, the sign of the cross product is determined by the direction of rotations of the vectors 
from U to V; from V to W  and  from W to U  in their common plane.


But since the sum of the vectors is zero,  U + V + W = 0,  the direction of rotation from U to V is the same 
as from V to W  and  the same as from V to W  in their common plane.


THEREFORE,  all three cross products are 3 equal vectors  U x V = V x W = W x U.

Answered and thoroughly explained.



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