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Addition of Vectors

If a+b+c=α ...

Question

If $a + b + c = α d, b + c + d = β a$ and a, b, c are non-coplanar vectors, then show the $a + b + c + d = 0$ .

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Solution

$\begin{matrix} S i n c e, \to a + \to b + \to c = α \to d a n d, \to b + \to c + \to d = β \to a T h i s i m p l i e s, \to a + \to b + \to c + \to d = (α + 1) \to d = (β + 1) \to a S i n c e, α \pm 1 \to d = (\frac{β + 1}{α + 1}) \to a \to a + \to b + \to c = α \to d = α (\frac{β + 1}{α + 1}) \to a (1 - \frac{α (β + 1)}{α + 1}) \to a + \to b + \to c = \to a T h i s i m p l i e s \to a + \to b + \to c a r e c o p l a n e r, W h i c h i s c o n t r a d i c t i o n H e n c e, α = - 1 \to a + \to b + \to c + \to d = \to 0 \end{matrix}$

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a, b, c

are three non-coplanar vectors such that

a + b + c = α d

b + c + d = β a

a + b + c + d

Q. If a, b, c are three non-coplanar vectors such that

a + b + c = α d

b + c + d = β a

a + b + c + d

\to a, \to b, \to c

are non-coplanar and

\to a + \to b + \to c = α \to d

\to b + \to c + \to d = β \to a

\to a + \to b + \to c + \to d =

a + b + c = α d a n d b + c + d = β a

and a,b,c are non-coplanar vectors , then show that

a + b + c + d = 0

\to a, \to b, \to c

are noncoplanar and

\to a + \to b + \to c = α \to d

\to b + \to c + \to d = β \to a

\to a + \to b + \to c + \to d =

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