User:Prof McCarthy/Net force

Parallelogram rule
A force is known as a bound vector which means it has a direction and magnitude and a point of application. A convenient way to define a force is by a line segment from a point A to a point B. If we denote the coordinates of these points as A=(Ax, Ay, Az) and B=(Bx, By, Bz), then the force vector applied at A is given by
 * $$ \mathbf{F}= \mathbf{B}-\mathbf{A} = (B_x-A_x, B_y-A_y, B_z-A_z). $$

The length of the vector B-A defines the magnitude of F, and is given by
 * $$ |\mathbf{F}| = \sqrt{(B_x-A_x)^2+(B_y-A_y)^2+(B_z-A_z)^2}. $$

The sum of two forces F1 and F2 applied at A can be computed from the sum of the segments that define them. Let F1=B-A and F2=C-A, then the sum of these two vectors is
 * $$ \mathbf{F}=\mathbf{F}_1+\mathbf{F}_2 = \mathbf{B}-\mathbf{A} + \mathbf{C}-\mathbf{A},$$

which can be written as
 * $$ \mathbf{F}=\mathbf{F}_1+\mathbf{F}_2 = 2(\frac{\mathbf{B}+\mathbf{C}}{2}-\mathbf{A})=2(\mathbf{V}-\mathbf{A}),$$

where V is the midpoint of the segment C-B that joins the points B and C.

Thus, the sum of the forces F1 and F2 is twice the segment joining A to the midpoint of the segment joining the endpoints of the two forces. The doubling of this length is easily achieved by defining a segments D-B and D-C parallel to C-A and C-A, respectively. The diagonal D-A of the parallelogram ABDC is the sum of the two vectors. This is known as the parallelogram rule.