Visualizing Vectors in Space

We can picture vectors in $\mathbb{R}^2$ or $\mathbb{R}^3$ in several ways. As a point: the vector $(3, 2)$ is the point 3 units right and 2 up. As an arrow: the same vector is drawn as an arrow from the origin to $(3, 2)$. As a free arrow: the same arrow shifted anywhere in the plane — direction and magnitude unchanged.

The arrow picture is powerful because it makes vector addition geometric: to add two vectors, place them tip-to-tail. The resulting arrow from start to end is the sum.

The magnitude (length) of $\mathbf{v} = (v_1, v_2)$ is $\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2}$ by the Pythagorean theorem. In $\mathbb{R}^m$: $\|\mathbf{v}\| = \sqrt{v_1^2 + \cdots + v_m^2}$.