Vectors can be combined in ways that preserve their geometric meaning.

Vector addition

If

$$\mathbf u=(u_1,\ldots,u_n),\qquad \mathbf v=(v_1,\ldots,v_n),$$

then their sum is obtained component by component:

$$\mathbf u+\mathbf v=(u_1+v_1,\ldots,u_n+v_n).$$

Geometrically, place the tail of $\mathbf v$ at the head of $\mathbf u$. The vector from the starting point to the final point is $\mathbf u+\mathbf v$.

For example,

$$(2,1)+(-1,3)=(1,4).$$

Scalar multiplication

Multiplying a vector by a real number $c$ multiplies every component:

$$c\mathbf v=(cv_1,\ldots,cv_n).$$

If $c>1$, the vector is stretched. If $0<c<1$, it is shortened. A negative scalar also reverses its direction. In particular,

$$-\mathbf v$$

has the same magnitude as $\mathbf v$ but points the opposite way.

Subtracting vectors

Vector subtraction is addition of the opposite:

$$\mathbf u-\mathbf v=\mathbf u+(-\mathbf v).$$

For points $A$ and $B$ represented by position vectors, the displacement from $A$ to $B$ is

$$\overrightarrow{AB}=\mathbf B-\mathbf A.$$

These operations make it possible to build new directions and displacements from known ones.

Visual intuition: vector operations