Parallelogram Law of Vector Addition
The parallelogram law of vector addition is a method that is used to find the sum of two vectors in vector theory. We study two laws for the addition of vectors  the triangle law of vector addition and the parallelogram law of vector addition. The parallelogram law of vector addition is used to add two vectors when the vectors that are to be added form the two adjacent sides of a parallelogram by joining the tails of the two vectors. Then, the sum of the two vectors is given by the diagonal of the parallelogram passing through the tail of the two vectors.
In this article, we will explore the parallelogram law of the addition of vectors, its formula, statement, and proof. We will learn to apply the law with the help of various examples for a better understanding of the concept.
What is Parallelogram Law of Vector Addition?
The parallelogram law of vector addition is the process of adding vectors geometrically. This law says, "Two vectors can be arranged as adjacent sides of a parallelogram such that their tails attach with each other and the sum of the two vectors is equal to the diagonal of the parallelogram whose tail is the same as the two vectors".
Consider the vectors P and Q in the figure below. To find their sum:
 Step 1: Draw the vectors P and Q such that their tails touch each other.
 Step 2: Complete the parallelogram by drawing the other two sides.
 Step 3: The diagonal of the parallelogram that has the same tail as the vectors P and Q represents the sum of the two vectors. i.e., P + Q = R.
☛Note: Here, the vector R is called the resultant vector (of P and Q).
Parallelogram Law of Vectors Formula
Consider two vectors P and Q with an angle θ between them. The sum of vectors P and Q is given by the vector R, the resultant sum vector using the parallelogram law of vector addition. If the resultant vector R makes an angle β with the vector P, then the formulas for its magnitude and direction are:
 R = √(P^{2} + Q^{2} + 2PQ cos θ)
 β = tan^{1}[(Q sin θ)/(P + Q cos θ)]
We will see the proof of these formulas in the section below.
Parallelogram Law of Vector Addition Proof
Let us first see the statement of the parallelogram law of vectors:
Statement of Parallelogram Law of Vector Addition: If two vectors can be represented by the two adjacent sides of a parallelogram drawn from a point, then their resultant sum vector is represented completely by the diagonal of the parallelogram drawn from the same point.
Now, to prove the formula of the parallelogram law, we consider two vectors P and Q represented by the two adjacent sides OB and OA of the parallelogram OBCA, respectively. The angle between the two vectors is θ. The sum of these two vectors is represented by the diagonal drawn from the same vertex O of the parallelogram, the resultant sum vector R which makes an angle β with the vector P.
Extend the vector P till D such that CD is perpendicular to OD. Since OB is parallel to AC, therefore the angle AOB is equal to the angle CAD as they are corresponding angles, i.e., angle CAD = θ. Now, first, we will derive the formula for the magnitude of the resultant vector R (side OC). Note that
 P = P
 Q = Q
 R = R
In rightangled triangle OCD, by Pythagoras theorem, we have
OC^{2} = OD^{2} + DC^{2}
⇒ OC^{2} = (OA + AD)^{2} + DC^{2}  (1)
In the right triangle CAD, we have
cos θ = AD/AC and sin θ = DC/AC
⇒ AD = AC cos θ and DC = AC sin θ
⇒ AD = Q cos θ and DC = Q sin θ  (2)
Substituting values from (2) in (1), we have
R^{2} = (P + Q cos θ)^{2} + (Q sin θ)^{2}
⇒ R^{2} = P^{2} + Q^{2}cos^{2}θ + 2PQ cos θ + Q^{2}sin^{2}θ
⇒ R^{2} = P^{2} + 2PQ cos θ + Q^{2}(cos^{2}θ + sin^{2}θ)
⇒ R^{2} = P^{2} + 2PQ cos θ + Q^{2} [cos^{2}θ + sin^{2}θ = 1]
⇒ R = √(P^{2} + 2PQ cos θ + Q^{2}) → Magnitude of the resultant vector R
Next, we will determine the direction of the resultant vector. We have in right traingle ODC,
tan β = DC/OD
⇒ tan β = Q sin θ/(OA + AD) [From (2)]
⇒ tan β = Q sin θ/(P + Q cos θ) [From (2)]
⇒ β = tan^{1}[(Q sin θ)/(P + Q cos θ)] → Direction of the resultant vector R
Some Special Cases of Parallelogram Law of Vector Addition
Now, we know the formula to determine the magnitude and direction of the sum of the two vectors. Let us consider a few special cases and substitute the values in the formula:
When the Two Vectors are Parallel (Same Direction)
If vectors P and Q are parallel, then we have θ = 0°. Substituting this in the formula of parallelogram law of vectors, we have
R = R = √(P^{2} + 2PQ cos 0 + Q^{2})
= √(P^{2} + 2PQ + Q^{2}) [Because cos 0 = 1]
= √(P + Q)^{2}
= P + Q
β = tan^{1}[(Q sin 0)/(P + Q cos 0)]
= tan^{1}[(0)/(P + Q cos 0)] [Because sin 0 = 0]
= 0°
When the Two Vectors are Acting in Opposite Direction
If vectors P and Q are acting in opposite directions, then we have θ = 180°. Substituting this in the formula of parallelogram law of vector addition, we have
R = √(P^{2} + 2PQ cos 180° + Q^{2})
= √(P^{2}  2PQ + Q^{2}) [Because cos 180° = 1]
= √(P  Q)^{2} or √(Q  P)^{2}
= P  Q or Q  P
β = tan^{1}[(Q sin 180°)/(P + Q cos 180°)]
= tan^{1}[(0)/(P + Q cos 0)] [Because sin 180° = 0]
= 0° or 180°
When the Two Vectors are Perpendicular
If vectors P and Q are perpendicular to each other, then we have θ = 90°. By parallelogram law of vector addition, we have
R = √(P^{2} + 2PQ cos 90° + Q^{2})
= √(P^{2} + 0 + Q^{2}) [Because cos 90° = 0]
= √(P^{2} + Q^{2})
β = tan^{1}[(Q sin 90°)/(P + Q cos 90°)]
= tan^{1}[Q/(P + 0)] [Because cos 90° = 0]
= tan^{1}(Q/P)
Important Notes on Parallelogram Law of Vector Addition:
 To apply the parallelogram law of vector addition, the two vectors are joined at the tails of each other and form the adjacent sides of a parallelogram.
 When the two vectors are parallel, then the magnitude of their resultant vector can be determined by simply adding the magnitudes of the two vectors.
 The triangle law and the parallelogram rule of vector addition are equivalent and give the same value as the resultant vector.
☛ Related Topics:
Parallelogram Law of Vector Addition Examples

Example 1: Which diagonal represents the sum of the two vectors p and q in the figure below?
Solution:
The tails of the vectors p and q join at O. Thus, the diagonal, which is the resultant of p and q also must start at O.
Thus, the resultant vector is OR.
Answer: p + q = OR.

Example 2: Two forces of magnitudes 4N and 7N act on a body and the angle between them is 45°. Determine the magnitude and direction of the resultant vector with the 4N force.
Solution:
Suppose vector P has magnitude 4N, vector Q has magnitude 7N and θ = 45°, then by the parallelogram law of vectors addition:
R = √(P^{2} + Q^{2} + 2PQ cos θ)
= √(4^{2} + 7^{2} + 2×4×7 cos 45°)
= √(16 + 49 + 56/√2)
= √(65 + 56/√2)
≈ 12.008 N
β = tan^{1}[(7 sin 45°)/(4 + 7 cos 45°)]
= tan^{1}[(7/√2)/(4 + 7/√2)]
≈ 28.95°
Answer: The magnitude is approximately 12 N and the direction is 28.95°.

Example 3:
Two vectors P = (1, 2) and Q = (2, 4) have an angle of 0° between them. Find the magnitude their sum vector.
Solution:
Using the parallelogram rule of vector addition formulas, we have
R = √(P^{2} + Q^{2} + 2PQ cos θ), β = tan^{1}[(Q sin θ)/(P + Q cos θ)]
For this, first, we need the magnitudes of vectors P and Q.
P = √(1^{2} + 2^{2}) = √5, Q = √(2^{2} + 4^{2}) = 2√5
R = √(√5^{2} + (2√5)^{2} + 2PQ cos θ)
= √(5 + 20 + 2×√5×2√5 cos 0°)
= √(25 + 20)
= √(45)
= 3 √5
Alternative Method:
Since the angle between the given two vectors is 0°, they must be parallel vectors. In that case,
R = P + Q = √5 + 2√5 = 3√5
Answer: The magnitude of the sum vector is 3√5 units.
FAQs on Parallelogram Law of Vector Addition
State Parallelogram Law of Vector Addition in Vector Algebra?
The parallelogram law of vector addition is used to add two vectors by making a parallelogram with the two vectors as adjacent sides (such that both of them have the same starting point). Then, the sum of the two vectors is given by the diagonal of the parallelogram starting at the same point as the two given vectors.
What is Parallelogram Law of Vectors Formula?
The sum of vectors P and Q is given by the vector R, the resultant sum vector. Let β be the angle between R and P, then the formulas for its magnitude and direction of R are:
 R = √(P^{2} + Q^{2} + 2PQ cos θ)
 β = tan^{1}[(Q sin θ)/(P + Q cos θ)]
When to Use Parallelogram Law of Vector Addition?
The parallelogram rule of vector addition is used when the sum of two vectors is to be determined. This law says the sum of two vectors is the diagonal of the parallelogram formed by the two vectors as adjacent sides.
How To Use Parallelogram Law of Vector Addition?
The parallelogram law says the sum of two vectors which when arranged as the adjacent sides of a parallelogram (and start at the same point) is the diagonal of the parallelogram that also starts at the same point.
How to Represent the Resultant Vector in Parallelogram Rule?
For any two vectors P and Q:
 The magnitude of the resultant vector R is, R = √(P^{2} + Q^{2} + 2PQ cos θ)
 The direction of R with P is, β = tan^{1}[(Q sin θ)/(P + Q cos θ)]
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