What is the parallelogram method

More about vectors

Vectors and scalars

Physical quantities that have both a direction and a magnitude (such as a force) become Vectors called.

Two force vectors acting at one point can be replaced by a single vector with the same effect. This force is called resultant or resultant for short. On the left you can see how to find the resultant in two simple cases. The search for the resultant of two or more vectors is called adding the vectors.

Physical quantities such as mass and volume, which have an amount but no direction, become Scalars called. Adding scalars is easy. A mass of 30 kg added to a mass of 40 kg always results in a mass of 70 kg.


Adding vectors: the parallelogram of forces



The parallelogram of forces is a geometric method to determine the resultant in situations like above, where the vectors are not in a line. It works like this:

To determine the resultant of two vectors (e.g. forces of 60 N and 80 N that act at a point O, as in the diagram below):

  1. Use the ruler to draw two red lines from point O to show the vectors (forces).
  2. Draw two black lines to draw a parallelogram.
  3. Draw a diagonal from O and determine its length with the ruler. The diagonal is the result of both size and direction.

Below is the result: The resulting force of both forces is 124 N.



You can also calculate the diagonal:

Resultant = $ \ mathrm {\ sqrt {(a + x) ^ 2 + h ^ 2}} $



We take the Pythagorean theorem as an aid and calculate x:

$ x ^ 2 + h ^ 2 = b ^ 2 \ \ big | \ - h ^ 2 $

$ x ^ 2 = b ^ 2 - h ^ 2 $

$ x ^ 2 = \ mathrm {(6 cm \ cdot 6cm) - (5 cm \ cdot 5cm) = 11 cm ^ 2} $

$ x = \ mathrm {\ sqrt {11cm ^ 2} = 3.317 cm} $

Now we can calculate the length of the diagonals according to the above equation:

$ \ sqrt {(a + x) ^ 2 + h ^ 2} = \ mathrm {\ sqrt {\ sqrt {(11.317 cm) ^ 2} + 25 cm ^ 2 = \ sqrt {153.07 cm ^ 2} = 12.37 cm} $


Or you can use trigonometry

The horizontal and vertical components of a force F can also be calculated using trigonometry:


In the green triangle above:

$ \ cos \ θ \ = \ \ frac {F_x} {F} $

and

$ \ sin \ θ \ = \ \ frac {F_y} {F} $

So:

$ F_x \ = \ F \ cos \ θ $

and

$ F_y \ = \ F \ sin θ $

The horizontal and vertical components of F are therefore as follows:



Components of a vector

The parallelogram method also works when you want to replace a single force with two forces. From a scientific point of view it is not a problem to split a vector into two components. If you use a parallelogram of forces, the diagonal describes the vector that you want to split into two components.



Above you can see some of the ways a force of 60N can be broken down into two components. There are endless other options.

Components at right angles

When working out the effects of a force, it sometimes helps to break the force down into components at right angles. For example, when a helicopter tilts its main rotor, the force has vertical and horizontal components that lift the helicopter and move it forward.



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