The Vandermonde determinant

Column operations. $C_2\leftarrow C_2-C_1$, $C_3\leftarrow C_3-C_1$:
$$V = \begin{vmatrix} 1 & 0 & 0 \\ a_1 & a_2-a_1 & a_3-a_1 \\ a_1^2 & a_2^2-a_1^2 & a_3^2-a_1^2 \end{vmatrix}.$$
Expand along the first row: $V = \begin{vmatrix} a_2-a_1 & a_3-a_1 \\ a_2^2-a_1^2 & a_3^2-a_1^2 \end{vmatrix}$.

Factorisation. $a_2^2-a_1^2=(a_2-a_1)(a_2+a_1)$ and similarly for $a_3$:
$$V = (a_2-a_1)(a_3-a_1)\begin{vmatrix} 1 & 1 \\ a_1+a_2 & a_1+a_3 \end{vmatrix} = (a_2-a_1)(a_3-a_1)(a_3-a_2).$$

Conclusion. $V(a_1,a_2,a_3)=(a_2-a_1)(a_3-a_1)(a_3-a_2)$. In particular, $V\neq0$ iff the $a_i$ are pairwise distinct.