Parallelogram Law

What is Parallelogram?

A Parallelogram is a four-sided quadrilateral whose opposite sides are parallel and congruent to each other. Opposite angles of a parallelogram are equal. The parallelogram and a rectangle are near about the same with one distinguishing property that the rectangle has all the angles of 900 and that of parallelogram does not.

In Mathematics, the parallelogram law belongs to elementary Geometry. This law is also known as parallelogram identity. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. In this article, let us study the definition of a parallelogram law, proof, and parallelogram law of vectors in detail.


Parallelogram Law of Addition

Parallelogram law of addition states that the sum of the squares of the length of the four sides of a parallelogram equals the sum of the squares of the length of the two diagonals. In Euclidean geometry, it is a must that the parallelogram should have equal opposite sides.

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If ABCD is a parallelogram, then AD = BC and AB = DC. Then according to the definition of the parallelogram law, it is stated as

2(AB)2 + 2(BC)2 = (AC)2 + (BD)2.

If a parallelogram is a rectangle, then the law is stated as

2(AB)2 + 2(BC)2 = 2(AC)2

Because in a rectangle, two diagonals are of equal lengths. i.e., (AC=BD)


Parallelogram Law of vectors

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If two vectors say vector p and vector q are acting simultaneously at a point, then it can be represented both in magnitude and direction by the adjacent sides drawn from a point. Therefore, the resultant vector is completely represented both in direction and magnitude by the diagonal of the parallelogram passing through the point.

Consider the above figure,

The vector P represents the side OA  and vector Q represents the side OB, respectively.

According to the parallelogram law, the side OC diagonal of the parallelogram represents the resultant vector R.

Vector OA + Vector OB = Vector OC

\[\overrightarrow{P}\]  + \[\overrightarrow{Q}\] = \[\overrightarrow{R}\]

 

Parallelogram Law of Addition of Vectors Procedure

Following are steps for the parallelogram law of addition of vectors are:

  • Draw a vector using a suitable scale in the direction of the vector

  • Draw the second vector using the same scale from the tail of the first vector

  • Treat these vectors as the adjacent sides and complete the parallelogram

  • Now, the diagonal represents the resultant vector in both magnitude and direction


Parallelogram Law Proof

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Step 1: Let AD=BC = p, AB = DC = q, and ∠ BAD = α

Step 2: Using the law of cosines in the BAD, we get

p2+ q2 – 2pqcos(α) = BD2 ——-(1)

Step 3: We know that in a parallelogram, the adjacent angles are supplementary so it sums up 1800. So

∠ADC = 180 – α

Step 4: Now, again use the law of cosines in the ADC

p2 + q2 – 2pqcos(180 – α) = AC2 ——-(2)

Step 5: Apply trigonometric identity cos(180 – x) = – cos x in step (2)

p2 + q2 + 2pqcos(α) = AC2

Step 6: Now, take the sum of the squares of the diagonals  adding equation 1 and 2

BD2 + AC2 =p2 + q2 – 2pqcos(α) + p2 + q2 + 2pqcos(α)

BD2 + AC2 =2p2 + 2q2 ——-(3)

BD2 + AC2 = 2(AB)2 + 2( BC)2

Hence, the parallelogram law is proved.

FAQ (Frequently Asked Questions)

1. What are Vectors?

A Physical quantity that has both magnitude and a direction is said to be a vector. In a line, the length of a line is a magnitude and the arrow is its direction. The start point is its tail and the endpoint is its head.

An increase and decrease in temperature is the best example of a vector, it has both magnitude and direction. There are ten types of vectors and they are 

  1. Zero vector

  2. Unit Vector

  3. Position Vector

  4. Co-initial Vector

  5. Like and Unlike Vectors

  6. Co-planar Vector

  7. Collinear Vector

  8. Equal Vector

  9. Displacement Vector

  10. Negative of a Vector

2. What is The Triangle Laws Of Vector Addition

Suppose, we have two vectors A and B as shown.

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To add this two vectors simply place the head of one vector over the tail of the other vector as shown.

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Now join the other end points of both the vectors together as shown,

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The result of the addition of given vectors is given by vector C which represents the sum of vectors  A and B.

i.e. C = A + B

Vector addition is commutative in nature i.e. C = A + B = B + A

Similarly if we have to subtract both the vectors using the triangle law then we simply reverse the direction of any vector and add it to another one as shown.

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Now we can mathematically represent this as

 C = A – B