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Question

If $$a, b, c$$ are three non-coplanar vectors such that $$a + b + c = \alpha d $$ and $$b + c + d = \beta a $$ then $$a+ b+ c+d $$ is equal to


A
0
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B
αa
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C
βb
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D
(α+β)c
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Solution

The correct option is A $$0$$
$$ a + b + c =\alpha (\beta a -b -c)$$
$$\Rightarrow  (1-\alpha\beta )a+(1+\alpha )b+(1+\alpha )c=0$$
$$\Rightarrow 1 =\alpha \beta $$ and $$ \alpha = -1 (a, b, c$$ are non-coplanar so linearly independent)
Substitute the value $$\alpha =-1,$$ we get $$a + b + c + d=0$$

Mathematics

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