Let a =2 i + j -2 k and b = i + j. Let c be a vector such that | c - a |=3, |( a × b ) × c |=3 and the angle between c and a × b be 30°. Then a · c…

Let $\displaystyle \overrightarrow{\mathrm{a}}=2 \hat{i}+\hat{j}-2 \hat{k}$ and $\displaystyle \overrightarrow{\mathrm{b}}=\hat{i}+\hat{j}$. Let $\displaystyle \vec{c}$ be a vector such that $\displaystyle |\vec{c}-\vec{a}|=3$, $\displaystyle |(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{c}}|=3$ and the angle between $\displaystyle \overrightarrow{\mathrm{c}}$ and $\displaystyle \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$ be $\displaystyle 30$°. Then $\displaystyle \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}$ is equal to :