1. The problem is to understand vector addition both algebraically and graphically. 2. Given vectors: $$\vec{a} = \begin{bmatrix} 6 \\ -2 \end{bmatrix}, \quad \vec{b} = \begin{bmatrix} -4 \\ 4 \end{bmatrix}$$ 3. Vector addition is done component-wise: $$\vec{a} + \vec{b} = \begin{bmatrix} 6 + (-4) \\ -2 + 4 \end{bmatrix} = \begin{bmatrix} 2 \\ 2 \end{bmatrix}$$ 4. Similarly, $$\vec{b} + \vec{a} = \begin{bmatrix} -4 + 6 \\ 4 + (-2) \end{bmatrix} = \begin{bmatrix} 2 \\ 2 \end{bmatrix}$$ 5. This shows vector addition is commutative: $$\vec{a} + \vec{b} = \vec{b} + \vec{a}$$ 6. Graphically, to add vectors, place the tail of the second vector at the head of the first vector. 7. For $$\vec{a} + \vec{b}$$, start at origin, draw $$\vec{a}$$ to point (6, -2), then from there draw $$\vec{b}$$ to point (2, 2). 8. For $$\vec{b} + \vec{a}$$, start at origin, draw $$\vec{b}$$ to point (-4, 4), then from there draw $$\vec{a}$$ to point (2, 2). 9. Both paths end at the same point (2, 2), confirming the algebraic result. 10. The graph visually demonstrates that vector addition is commutative because the resultant vector is the same regardless of the order. This is why the graph shows two different paths but the same final vector.