To add two vectors ** A** y

**forming an angle to each other, two methods are used: the triangle method and the**

*B***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} ] = (L_{1})^{2 }+ (L_{2})^{2}

If the parallelogram is a rectangle, the diagonals L_{1} and L_{2} they are equal, therefore, the parallelogram law is reduced to the Pythagorean theorem.

2 [(AB)^{2 }+ (CD)^{2} ] = 2 (L_{1})^{2}

[(AB)^{2 }+ (CD)^{2} ] = 2 (L_{1})^{2}

**Demonstration**

Let ABCD be a parallelogram whose diagonals are L_{1} and L_{2} . 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 L_{1} and L_{2}:

L_{1 }= √ [(x + x_{1})^{2} + and^{2} ] = √ [x^{2} + 2xx_{1} + (x_{1})^{2} + and^{2} ]

L_{2 }= √ [(x_{1 } - x)^{2} + and^{2} ] = √ [(x_{1})^{2} - 2xx_{1} + x^{2} + and^{2} ]

Thus:

(L_{1})^{2 }+ (L_{2})^{2} = x^{2} + 2xx_{1} + (x_{1})^{2} + and^{2 }+ [(x_{1})^{2} - 2xx_{1} + x^{2} + and^{2} =

= 2 [x^{2} + (x_{1})^{2} + and^{2}]

Calculating the length of AB and CD, we have:

AB = x

CD = √ [(x_{1})^{2 }+ and^{2} ]

Now:

2 [(AB)^{2 }+ (CD)^{2} ] = 2 [x^{2} + (x_{1})^{2} + and^{2}]

Thus:

[(AB)^{2 }+ (CD)^{2} ] = 2 (L_{1})^{2}

**Vector parallelogram law or method**

To add vectors ** A** y

**by the parallelogram method:**

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

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

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

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