Vector Addition

To add two vectors, add their components:

$$\mathbf{u} + \mathbf{v} = \begin{pmatrix} u_1 + v_1 \\ u_2 + v_2 \end{pmatrix}$$

Geometric Meaning

Geometrically, adding vectors means placing them head-to-tail. The result is the vector from the start of the first to the end of the last.

Properties

• Commutative: $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$
• Associative: $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$
• Zero vector: $\mathbf{v} + \mathbf{0} = \mathbf{v}$