**Parallelogram Law of Vector Addition**

**Parallelogram Law of Vector Addition**: It is the Parallelogram Law of Vector Addition is a technique that can be used to calculate the total of two vectors in the theory of vectors. We look at two laws that govern an addition of vectors: that of the triangular law for vector addition as well as the law of parallelograms for vector addition. A parallelogram-based law for vector addition can be utilized to add two vectors if the vectors that need to be added are the two sides of a parallelogram , by connecting the tails both vectors. Then the sum of these two vectors can be calculated using its diagonal.

This article we’ll look at the parallelogram law, the addition of vectors, as well as its formula, its statement, and its the proof. We will be taught how to apply this law through the use of several examples to gain an understanding of the idea.

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**What is Parallelogram Law of Vector Addition?**

A sum between two vectors is determined through the vector addition method and the Parallelogram Law of Vector Addition is an rule that allows you to find the sum vector. If a fish is moving across the water from the opposite part of the river using the vector Q, and the water flowing in the river is flowing in a direction that is parallel to the vector P, as illustrated in the image below.

Now, the net speed for the fish is the product of the two velocity levels that is, the speed of the fish as well as the speed of the flow of the river. This will have a different speed. This means that the fish moves on a different path, that is the sum of the two velocity. To determine how fast the fish is moving, we could take the two vectors to be opposite sides of the parallelogram and apply the law of parallelograms for vector addition to find the sum vector.

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**Parallelogram Law of Vector Addition Formula**

Take the two vectors P and Q that have an angle of th. The sum of the vectors P and Q can be calculated via the vector. It is the resultant sum vector utilizing the law of parallelograms in vector addition. If the resultant R is an angle **with the vector ph** by combining the vector P the formulas for the magnitude and direction of the angle are:

- |R| = (P
^{2}+ Q^{2}+ 2PQ cos th) - B = Tan
^{b = tan}[(Q sin th)/(P + Q cos th)[]

**Parallelogram Law of Vector Addition Proof**

We will first examine the explanation in the law of parallelograms vector addition:

**Declaration of the Parallelogram Law on Vector Addition** If two vectors are represented by the two adjoining sides (both in direction and magnitude) of the parallelogram drawn from a single point, their sum vector is completely represented through the diagonals of the Parallelogram drawn from that same point.

To prove the mathematical formula behind the parallelogram law, take two vectors P and Q, which are represented by the two edges OB as well as OA in the Parallelogram OBCA as well as. The angle between these two vectors is. This sum can be depicted with the help of the diagonal drawn by using the the vertex O on the diagonalogram. the sum vector R that creates an angle b when combined with this vector.

The vector P should be extended until D, so the CD runs parallel the direction of. Because OB runs parallel to AC the angle AOB will be equal with the angle of CAD since they are both corresponding angles i.e. angle CAD is equal to the. Then we’ll find the formula to calculate how much of the vector R (side OC).

In the right-angled triangle OCD we can see

OC^{2} = OD^{2} + DC^{2}

= OC^{2} = (OA + AD)^{2} + DC^{2} — (1)

In the right-hand triangle CAD We have

COS TH = AD/AC, and sin the = DC/AC

= AD = AC cos th and DC = AC sin th

= AD = Q cos th and DC = Q sin th — (2)

By substituting the values of (2) (1). In (1) We have

2. ^{2} = (P + Q cos the) ^{2} + (Q sin the) ^{2}

= R2 = P2 + Q2cos2th + 2PQ cos th + Q2sin2th

= R2 = P2 + 2PQ cos th + Q2(cos2th + sin2th)

= R2 = P2 + 2PQ cos th + Q2 [cos2th + sin2th = 1]

= R = (P ^{2} + 2PQ cos 2 + Q ^{2}) The Magnitude of the vector that results R

The next step is to identify the direction for the resulting vector. We are in the correct traingle ODC,

tan b = DC/OD

= tan b = Q sin th/(OA + AD) [From (2)]

= Tan B = Q sin th/(P + Q cos th) [From (2)= tan b = Q sin th/(P + Q cos

= b = Tan ^{= b = tan}[(Q sin th)/(P + Q cos th)(P + Q cos th)] – Direction of the resulting vector R

**Some Special Cases of Parallelogram Law of Vector Addition**

Now, we have the formula used to calculate how much and the direction to be found in the total of two vectors. Let’s look at the following special cases, and then substutte the formula: the formula:

**When the Two Vectors are Parallel (Same Direction)**

If the vectors P and Q are both parallel each other, we get Th = 0deg. By substituting this formula in the formula of the Parallelogram law of vector Addition We have

|R| = (P^{2} + 2PQ cos 0 + Q^{2})

= (P^{2} + 2PQ + Q^{2}) [Because cos 0 = 1]

= (P + Q)^{2}

= P + Q

b = tan^{-1}[(Q sin 0)/(P + Q cos 0)]

= tan^{-1}[(0)/(P + Q cos 0)] [Because sin 0 = 0]

= 0deg

**When the Two Vectors are Acting in Opposite Direction**

If the vectors P and Q are operating in opposite directions, we will have the angle of 180 degrees. In the formula of the Parallelogram law of vector Addition and we get

|R| = (P^{2} + 2PQ cos 180deg + Q^{2})

= (P^{2} – 2PQ + Q^{2}) [Because cos 180deg = -1]

= (P – Q) ^{2} or (Q – P) ^{2}

= P – Q or Q – P

b = tan^{-1}[(Q sin 180deg)/(P + Q cos 180deg)]

= tan^{-1}[(0)/(P + Q cos 0)] [Because sin 180deg = 0]

= 0deg or 180deg

**When the Two Vectors are Perpendicular**

If vectors Q and P are perpendicular each other and we find the ratio th = 90deg. By substituting this into the formula of the Parallelogram law of vector Addition and we get

|R| = (P^{2} + 2PQ cos 90deg + Q^{2})

= (P^{2} + 0 + Q^{2}) [Because cos 90deg = 0]

= (P^{2} + Q^{2})

b = tan^{-1}[(Q sin 90deg)/(P + Q cos 90deg)]

= tan^{-1}[Q/(P + 0)] [Because cos 90deg = 0]

= tan^{-1}(Q/P)

**Important Notes on Parallelogram Law of Vector Addition**

- To implement to apply the Parallelogram Law of Vector Addition The two vectors are joined at the tails of the other , and create the adjacent sides of the parallelogram.
- If the two vectors are paired and parallel, their magnitude of the resulting vector can be calculated by addition of the values for both vectors.
- The triangle law as well as the law of parallelograms for addition of vectors are identical and provide the same amount in the form of a vector.

Parallelogram Law of Vector Addition Examples

**Example 1:** Two forces of magnitudes of 7N and 4N exert force on a person and the angle between them is 45 degrees. Find the direction and magnitude of the resulting vector using the force of 4N using the Parallelogram Law of Vector Addition.

**Answer:** Suppose vector P is magnitude 4N and vector Q’s magnitude is 7N, and the angle is 45deg. we have the formulas

|R| = (P^{2} + Q^{2} + 2PQ cos th)

= (4^{2} + 7^{2} + 2x4x7 cos 45deg)

= (16 + 49 + 56/2)

= (65 + 56/2)

= 12.008 N

b = tan^{-1}[(7 sin 45deg)/(4 + 7 cos 45deg)]

= tan^{-1}[(7/2)/(4 + 7/2)]

= tan^{-1}[(7)/(42 + 7)]

**Answer** Magnitude of 12N, and the orientation is ^{1}[(7)/(42 + 7)[7, 42].