# Parallelogram law

To add two vectors A y B forming an angle to each other, two methods are used: the triangle method and the parallelogram method. The parallelogram method is the most widely used method; in it the two vectors are drawn at the origin of a Cartesian plane, respecting their magnitudes, directions and directions. The resulting vector will be the diagonal of the parallelogram starting at the origin of the Cartesian plane (Figure I).

## Parallelogram law

According to parallelogram law, the sum of the squares of the four sides of a parallelogram is equal to the sum of the squares of the two diagonals of it, that is:

2 [(AB)2 + (CD)2 ] = (L1)2 + (L2)2

If the parallelogram is a rectangle, the diagonals L1 and L2 they are equal, therefore, the parallelogram law is reduced to the Pythagorean theorem.

2 [(AB)2 + (CD)2 ] = 2 (L1)2

[(AB)2 + (CD)2 ] = 2 (L1)2

### Demonstration

Let ABCD be a parallelogram whose diagonals are L1 and L2 . it lies at the origin of a Cartesian coordinate system.

Suppose AB = CD and AD = BC. From the distance formula we can determine the length of L1 and L2:

L1 = √ [(x + x1)2 + and2 ] = √ [x2 + 2xx1 + (x1)2 + and2 ]

L2 = √ [(x1  - x)2 + and2 ] = √ [(x1)2 - 2xx1 + x2 + and2 ]

Thus:

(L1)2 + (L2)2 = x2 + 2xx1 + (x1)2 + and2 + [(x1)2 - 2xx1 + x2 + and2 =

= 2 [x2 + (x1)2 + and2]

Calculating the length of AB and CD, we have:

AB = x

CD = √ [(x1)2 + and2 ]

Now:

2 [(AB)2 + (CD)2 ] = 2 [x2 + (x1)2 + and2]

Thus:

[(AB)2 + (CD)2 ] = 2 (L1)2

## Vector parallelogram law or method

To add vectors A y B by the parallelogram method:

1. We draw the vector A at the origin of a Cartesian plane respecting its module, direction and direction.

1. We draw on the head of A, the vector B respecting its module, direction and sense.

1. Lines are drawn parallel to each vector forming a parallelogram.

1. The resulting vector will be the diagonal of the parallelogram starting at the origin of the Cartesian plane.