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=(A_x, A_y, A_z) and B=(B_x, B_y, B_z), 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.