Let me analyze the content of the image and provide you with a solution to the mathematical problem.The exercise provided involves calculations based on vectors $\mathbf{u} = 2\mathbf{i}$ and $\mathbf{v} = -\frac{1}{2}\mathbf{i} + \frac{\sqrt{3}}{2}\mathbf{j}$ in an orthonormal basis $(\mathbf{i}, \mathbf{j})$. Let's break it down:

1a. Calculate $(\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} - \mathbf{v})$

Step 1: Add and subtract vectors

[ \mathbf{u} + \mathbf{v} = (2 - \frac{1}{2})\mathbf{i} + \frac{\sqrt{3}}{2}\mathbf{j} = \frac{4}{2} - \frac{1}{2}\mathbf{i} + \frac{\sqrt{3}}{2}\mathbf{j} \mathbf{}). <