Let the vectors a =(1+ t ) i +(1- t ) j + k, b =(1- t ) i +(1+ t ) j +2 k and c = t i - t j + k, t ∈ R be such that for α, β, γ ∈ R, α a +β b +γ c =…

Let the vectors $\displaystyle \overrightarrow{\mathrm{a}}=(1+\mathrm{t}) \hat{i}+(1-\mathrm{t}) \hat{j}+\hat{k}, \overrightarrow{\mathrm{~b}}=(1-\mathrm{t}) \hat{i}+(1+\mathrm{t}) \hat{j}+2 \hat{k}$ and $\displaystyle \overrightarrow{\mathrm{c}}=\mathrm{t} \hat{i}-\mathrm{t} \hat{j}+\hat{k}, \mathrm{t} \in \mathbf{R}$ be such that for $\displaystyle \alpha, \beta, \gamma \in \mathbf{R}, \alpha \overrightarrow{\mathrm{a}}+\beta \overrightarrow{\mathrm{b}}+\gamma \overrightarrow{\mathrm{c}}=\overrightarrow{0} \Rightarrow \alpha=\beta=\gamma=0$. Then, the set of all values of t is :