Determine the value of A · ( A × B ) .

The cross product $\vec{A} \times \vec{B}$ yields a vector that is perpendicular to both $\vec{A}$ and $\vec{B}$.

Let $\vec{C} = \vec{A} \times \vec{B}$. Since $\vec{C}$ is perpendicular to $\vec{A}$, the angle between them is $90^{{}^{{}^{\circ}}}$.

The dot product $\vec{A} \cdot (\vec{A} \times \vec{B}) = \vec{A} \cdot \vec{C} = |\vec{A}| |\vec{C}| \cos 90^{{}^{{}^{\circ}}} = 0$.

Alternatively, the expression represents the scalar triple product $[\vec{A} \quad \vec{A} \quad \vec{B}]$, which is zero because two of the vectors are identical.