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If a+b,b+c,c+a are three non coplanar vectors then prove that axb,b+c,c+a are also non coplanar.

If a+b,b+c,c+a are three non coplanar vectors then prove that axb,b+c,c+a are also non coplanar.

Grade:12th pass

1 Answers

Vikas TU
14149 Points
4 years ago
lets give a name to these new vectors:
u⃗ =a⃗ +b⃗ 
v⃗ =a⃗ +c⃗ 
w⃗ =b⃗ +c⃗ 
then it is obvious that:
2a⃗ =u⃗ +v⃗ −w⃗ 
2b⃗ =u⃗ +w⃗ −v⃗ 
2a⃗ =v⃗ +w⃗ −u⃗ 
as you see we can get our three non-coplanar vectors back, so the new ones are non-coplanar.
i could also have said that det of
1 1 0
1 0 1  = -2 and is not equal to zero.
0 1 1 

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