In a parallelogram $$ABCD$$, $$\left|\overrightarrow{AB}\right| = a$$, $$\left|\overrightarrow{AD}\right| = b$$ and $$\left|\overrightarrow{AC}\right| = c$$. $$\overrightarrow{DB} \cdot \overrightarrow{AB}$$ has the value:

$$\vec{AC} = \vec{AB} + \vec{AD} = \mathbf{a} + \mathbf{b}$$

$$\vec{DB} = \vec{AB} - \vec{AD} = \mathbf{a} - \mathbf{b}$$

$$c^2 = |\vec{AC}|^2 = |\mathbf{a} + \mathbf{b}|^2$$

$$c^2 = \mathbf{a} \cdot \mathbf{a} + \mathbf{b} \cdot \mathbf{b} + 2(\mathbf{a} \cdot \mathbf{b})$$

$$c^2 = a^2 + b^2 + 2(\mathbf{a} \cdot \mathbf{b})$$

$$\mathbf{a} \cdot \mathbf{b} = \frac{c^2 - a^2 - b^2}{2}$$

$$\vec{DB} \cdot \vec{AB} = (\mathbf{a} - \mathbf{b}) \cdot \mathbf{a}$$

$$\vec{DB} \cdot \vec{AB} = (\mathbf{a} \cdot \mathbf{a}) - (\mathbf{b} \cdot \mathbf{a})$$

$$\vec{DB} \cdot \vec{AB} = a^2 - (\mathbf{a} \cdot \mathbf{b})$$

$$\vec{DB} \cdot \vec{AB} = a^2 - \left( \frac{c^2 - a^2 - b^2}{2} \right)$$

$$\vec{DB} \cdot \vec{AB} = \frac{3a^2 + b^2 - c^2}{2}$$