Addition and Subtraction of Vectors — Graphical Method

Because vectors carry direction as well as magnitude, they cannot, in general, be added by

simply adding their magnitudes. Instead, vector addition follows the triangle law: to

add $\vec A$ and $\vec B$, draw $\vec A$ first, then draw $\vec B$ starting from the head

(tip) of $\vec A$; the resultant vector $\vec R = \vec A + \vec B$ is the vector drawn from

the tail of $\vec A$ to the head of $\vec B$, closing the triangle. Equivalently, the

parallelogram law places $\vec A$ and $\vec B$ tail-to-tail as two adjacent sides of a

parallelogram; the resultant $\vec R$ is then the diagonal of the parallelogram drawn from

the common tail. Both constructions give the identical resultant vector (see the figure).

If $\theta$ is the angle between $\vec A$ and $\vec B$, the law of cosines applied to the

triangle gives the magnitude of the resultant:

The angle $\alpha$ that $\vec R$ makes with $\vec A$ follows from the sine rule:

Two special cases are worth remembering: when $\vec A$ and $\vec B$ point in the same

direction ($\theta = 0^\circ$), the resultant is simply $R = A + B$, the algebraic sum; when

they point in exactly opposite directions ($\theta = 180^\circ$), the resultant is

$R = |A - B|$, the algebraic difference. For any other angle, the resultant magnitude lies

strictly between $|A-B|$ and $A+B$.

Vector addition is both commutative, $\vec A + \vec B = \vec B + \vec A$ (the

parallelogram construction is symmetric in $\vec A$ and $\vec B$), and associative,

$(\vec A + \vec B) + \vec C = \vec A + (\vec B + \vec C)$ (adding three or more vectors by

repeated application of the triangle law, in any grouping or order, gives the same final …