A scalar is any positive or negative physical quantity that can be completely specified by its magnitude. Examples of scalar quantities include length, mass, and time.
A vector is any physical quantity that requires both a magnitude and a direction for its complete description. Examples of vectors encountered in statics are force, position, and moment.
A vector is shown graphically by an arrow. The length of the arrow represents the magnitude of the vector, and the angle $\theta$ between the vector and a fixed axis defines the direction of its line of action. The head or tip of the arrow indicates the sense of direction of the vector.
In print, vector quantities are represented by boldface letters such as $\mathbf{A}$, and the magnitude of a vector is italicized, A. For handwritten work, it is often convenient to denote a vector quantity by simply drawing an arrow above it, $\vec{A}$.
If a vector is multiplied by a positive scalar, its magnitude is increased by that amount. Multiplying by a negative scalar will also change the directional sense of the vector.
Vector Addition. When adding two vectors together it is important to account for both their magnitudes and their directions. To do this we must use the parallelogram law of addition. To illustrate, the two component vectors $\mathbf{A}$ and $\mathbf{B}$ are added to form a resultant vector $\mathbf{R} = \mathbf{A} + \mathbf{B}$ using the following procedure:
We can also add $\mathbf{B}$ to $\mathbf{A}$ using the triangle rule, which is a special case of the parallelogram law, whereby vector $\mathbf{B}$ is added to vector $\mathbf{A}$ in a "head-to-tail" fashion by connecting the head of $\mathbf{A}$ to the tail of $\mathbf{B}$. The resultant $\mathbf{R}$ extends from the tail of $\mathbf{A}$ to the head of $\mathbf{B}$.
In a similar manner, $\mathbf{R}$ can also be obtained by adding $\mathbf{A}$ to $\mathbf{B}$. By comparison, it is seen that vector addition is commutative; in other words, the vectors can be added in either order, i.e.,
$$ \mathbf{R} = \mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A} $$
As a special case, if the two vectors $\mathbf{A}$ and $\mathbf{B}$ are collinear, both have the same line of action, the parallelogram law reduces to an algebraic or scalar addition $R = A + B$.