Parallelogram law

parallelogram method
Figure 2. Parametric equations of the curve

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

parallelogram lawIf 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.

parallelogram law

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.

parallelogram law

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

parallelogram law

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

parallelogram law

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

parallelogram law