If bar{a}, bar{b}, bar{c} are non-zero vector | vedclass

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(D) Given that $\bar{a} + \bar{b}$ is parallel to $\bar{c}$,there exists a scalar $k \in \mathbb{R}$ such that $\bar{a} + \bar{b} = k\bar{c}$  $(1)$Al

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$\bar{0}$

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(D) Given that $\bar{a} + \bar{b}$ is parallel to $\bar{c}$,there exists a scalar $k \in \mathbb{R}$ such that $\bar{a} + \bar{b} = k\bar{c}$  $(1)$Also,$\bar{b} + \bar{c}$ is parallel to $\bar{a}$,so there exists a scalar $m \in \mathbb{R}$ such that $\bar{b} + \bar{c} = m\bar{a}$  $(2)$From $(2)$,$\bar{c} = m\bar{a} - \bar{b}$. Substituting this into $(1)$:$\bar{a} + \bar{b} = k(m\bar{a} - \bar{b})$$\bar{a} + \bar{b} = km\bar{a} - k\bar{b}$$(1 - km)\bar{a} + (1 + k)\bar{b} = \bar{0}$Since $\bar{a}$ and $\bar{b}$ are non-zero and not parallel,their coefficients must be zero:$1 - km = 0 \implies km = 1$$1 + k = 0 \implies k = -1$Substituting $k = -1$ into $km = 1$,we get $m = -1$.Now,substituting $k = -1$ into $(1)$:$\bar{a} + \bar{b} = -1\bar{c} \implies \bar{a} + \bar{b} + \bar{c} = \bar{0}$

If $\bar{a}, \bar{b}, \bar{c}$ are non-zero vectors such that no two of them are parallel,and $\bar{a} + \bar{b}$ is parallel to $\bar{c}$,and $\bar{b} + \bar{c}$ is parallel to $\bar{a}$,then $\bar{a} + \bar{b} + \bar{c} = $

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