Given the equilateral triangle $ABC$, I know $A(0,0,0)$ and I'm trying to find the coordinates of $B$ and $D$.
I know $$AB\cdot AC=8$$ and the normalized vectors associated to them are $$AB_0=(1,0,0)$$ and $$AC_0=\left(\frac{1}{2},\frac{\sqrt{3}}{2},0\right)$$
What I did was: since $ABC$ is equilateral, then $$||AB||=||AC||$$ so $$||AB\cdot AC||=||8||\Rightarrow||AB||\cdot||AC||=8$$ Therefore, their norms are both equal to $\sqrt{8}$.
Hence, there are real numbers $\alpha$ and $\beta$ such that the vectors $$\alpha(1,0,0),\qquad\beta\left(\frac{1}{2},\frac{\sqrt{3}}{2},0\right)$$ have norms $\sqrt{8}$, which clearly gives me $$\alpha=\beta=\sqrt{8}$$
But when I try to do the dot product between $$(\sqrt{8},0,0)$$ and $$\left(\frac{\sqrt{8}}{2},\frac{\sqrt{8}\cdot\sqrt{3}}{2},0\right)$$ I get 4 as result. What is wrong?