Parallelogram Law: Overview, Questions, Preparation

In mathematics, parallelogram law is the fundamental law of elementary geometry. This law is also known as the identity of a parallelogram. In this article, let us look in-depth at the definition of parallelogram law, proof law, and parallelogram law of vectors. 

Parallelogram Law of Addition

The parallelogram rule specifies that the sum of the squares of the length of the four sides of the parallelogram is equal to the sum of the squares of the length of the two diagonals. In Euclidean geometry, the parallelogram can have the same opposite faces.

If ABCD is a parallelogram, then AB is DC and AD is BC. Then, according to the meaning of the parallelogram rule, it shall be specified as 

2(AB)²+ 2 (BC)² = (AC)² + (BD)²

If the parallelogram is a rectangle, the rule shall be as follows: 

2(AB)²+ 2 (BC)² = 2(AC)²

That is a rectangle, and two diagonals are of similar length. That is, (AC = BD)

Parallelogram Law on the addition of vectors method 

• The measures for the parallelogram rule on the addition of vectors are as follows: 
• Draw a vector using the appropriate scale in the vector direction. 
• Draw the second vector using the same scale from the tail of the first vector. 
• Use these vectors as opposite sides to complete the parallelogram. 
• Now, the diagonal reflects the resulting vector in both magnitude and direction.

Let AD=BC = s, AB = DC = r, and ∠ BAD = α

Using the law of cosines in the triangle BAD, we get

s² + r² – 2sr cos(α) = BD²               ——-(1)

We know that in a parallelogram, the adjacent angles are supplementary. So

∠ADC = 180 – α

Now, again use the law of cosines in the triangle ADC

s² + r²– 2sr cos(180 – α) = AC² ——-(2)

Apply trigonometric identity cos(180 – x) = – cos s in (2)

s² + r²+ 2sr cos(α) = AC²

Now, the sum of the squares of the diagonals (BD2 + AC2) are represented as,

BD² + AC² = s² + r² – 2srcos(α) + s² + r² + 2sr cos(α)

Simplify the above expression, we get

BD² + AC² =2x² + 2r²                 ——-(3)

The above equation is represented as:

BD² + AC² = 2(AB)² + 2(BC)²

Hence, the parallelogram law is proved.

This concept is taught under chapter vector algebra. You will learn about the law and its application in algebra. The weightage is 4 marks in the final exam.

1. Let (a→= 3 î + 4 ĵ – 7 k̂) and (b→= 6 î + 4 ĵ – 6 k̂). Add the two vectors.

Solution.  Because both vectors are already represented in a coordinated structure, we may add both explicitly as follows. 

→    → 

a  +  b = (3 + 6) î + (4 + 4) ĵ + (−7 – 6) k̂

or

→  → 

a + b = 9 î + 8 ĵ − 13 k̂

Q: What's the parallelogram law of the forces? 

Q: How can you assert that the parallelogram is the law? 

Q: What is the concept of a parallelogram? 

Q: Are the opposite angles of a parallelogram equal? 

Q: What is a parallelogram?