If $¯ a, ¯ b$ and $¯ c$ are three non-coplaner vectors then vector $¯ r$ in space can be expreesed as a linear combination of $¯ a, ¯ b, ¯ c$ ?

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Solution

If $\to a, \to b, \to c$ be three non zero, non coplanar vectors in space, then any vector $\to r$ can be expressed uniquely as a linear combination of $\to a, \to b, \to c$ i.e. there exists $l, m, n \in R$ such that $l \to a + m \to b + n \to c = 0$
This also means that if $l_{1} \to a + m_{1} \to b + n_{1} \to c = l_{2} \to a + m_{2} \to b + n_{2} \to c$ then $l_{1} = l_{2}, m_{1} = m_{2} a n d n_{1} = n_{2}$

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