Quick questions on Vector arithmetic: addition, scalar multiplication, magnitude and unit vectors

A vector carries two pieces of information at once, a magnitude (length) and a direction, and nothing else. Picture it as an arrow. Crucially, a vector is free: an arrow of the same length pointing the same way is the same vector, wherever you draw it on the page. This is why addition works by sliding arrows around, and why $\overrightarrow{AB}$ depends only on where $A$ and $B$ are, not on any axes.

To add vectors, add components:

For a scalar $\lambda$ and a vector $\mathbf{a}$,

The magnitude (length, or modulus) of $\mathbf{a}$ is

A unit vector has magnitude $1$. The unit vector in the direction of a non-zero $\mathbf{a}$ is