$\overrightarrow r$ is a non zero vector, but it is given $\overrightarrow r.\overrightarrow a=0 \: and \: \overrightarrow r.\overrightarrow b=0\;and\;\overrightarrow r.\overrightarrow c=0$
This implies $\overrightarrow a.\overrightarrow b$ and $\overrightarrow c$ should be zero
Hence $ \overrightarrow a.(\overrightarrow b \times \overrightarrow c)=0$
Therefore If $\overrightarrow{r}.\overrightarrow{a}=0,\overrightarrow{r}.\overrightarrow{b}=0$ and $\overrightarrow{r}.\overrightarrow{c}=0$ for some non-zero vector $\overrightarrow{r}$,then the value of $\overrightarrow{a}.(\overrightarrow{b}\times\overrightarrow{c})$ is zero