What is Parallelogram?
A Parallelogram is a four-sided quadrilateral whose opposite sides are parallel and congruent to each other. The 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 90°0 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
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
→ → →
P + Q = R
Parallelogram Law of Addition of Vectors Procedure
Following are the steps for the parallelogram law of addition of vectors:
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 180°0. 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 equations 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.
FAQs on Parallelogram Law
1. What is the Parallelogram Law of Vector Addition?
The Parallelogram Law of Vector Addition states that if two vectors acting at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram, then their resultant vector is represented in both magnitude and direction by the diagonal of the parallelogram passing through that same point.
2. What is the formula to find the magnitude of the resultant vector in the Parallelogram Law?
To find the magnitude of the resultant vector (R) when two vectors P and Q are inclined at an angle θ, you can use the following formula derived from the law:
R = √(P² + Q² + 2PQ cosθ)
This formula is a direct application of the Law of Cosines to the triangle formed by the vectors and their resultant.
3. How does the Parallelogram Law differ from the Triangle Law of Vector Addition?
While both laws yield the same result, their conceptual approach and graphical representation differ:
- Vector Arrangement: In the Parallelogram Law, vectors are arranged tail-to-tail (co-initial), representing two forces acting from a single point. In the Triangle Law, vectors are arranged head-to-tail.
- Resultant Representation: In the Parallelogram Law, the resultant is the diagonal from the common starting point. In the Triangle Law, the resultant is the third side that closes the triangle, drawn from the tail of the first vector to the head of the second.
4. What do the two diagonals of the parallelogram represent in vector addition?
The two diagonals of the parallelogram represent different vector operations. The diagonal that originates from the common point of the two vectors (P and Q) represents their sum or resultant (P + Q). The other diagonal, which connects the heads of the two vectors, represents the difference between the vectors (P - Q or Q - P).
5. Can the Parallelogram Law be used for vector subtraction?
Yes, the Parallelogram Law can be adapted to perform vector subtraction. Subtracting a vector Q from a vector P (i.e., P - Q) is equivalent to adding P and the negative of Q (i.e., P + (-Q)). To do this, you would construct a parallelogram using vector P and vector -Q (which has the same magnitude as Q but the opposite direction). The main diagonal of this new parallelogram would represent the resultant of the subtraction.
6. How is the Parallelogram Law applied to physical quantities like forces?
The Parallelogram Law is fundamental in physics, especially in mechanics. For example, if two forces, F1 and F2, are pulling an object from the same point in different directions, they can be represented as the adjacent sides of a parallelogram. The diagonal of this parallelogram gives the net force (the resultant force), indicating the final direction and magnitude of the object's acceleration as if it were acted upon by a single force.
7. What is the main condition for applying the Parallelogram Law of vectors?
The primary condition for applying the Parallelogram Law is that the two vectors must be co-initial. This means their tails must start from the same point. The law is specifically designed to find the resultant of vectors that act simultaneously on a single point in space, such as two forces acting on a body or the combined velocity of an object.
