Triangle Law of Vector Addition
Triangle Law of Vector Addition is a mathematical concept that is used to find the sum of two vectors. This law is used to add two vectors when the first vector's head is joined to the tail of the second vector and then joining the tail of the first vector to the head of the second vector to form a triangle, and hence obtain the resultant sum vector. That's why the triangle law of vector addition is also called the headtotail method for the addition of vectors.
Let us study the triangle law of the addition of vectors, its statement, formula, and proof. This law is used to determine the net displacement, velocity, acceleration, etc. We will also solve questions and examples based on the triangle law of vector addition to understand its application and the concept.
1.  What is the Triangle Law of Vector Addition? 
2.  Triangle Law of Vector Addition Formula 
3.  Triangle Law of Vector Addition Proof 
4.  FAQs on Triangle Law of Vector Addition 
What is the Triangle Law of Vector Addition?
The triangle law of vector addition is a law that is used in vector algebra to determine the resultant sum vector when two or more vectors are added. Suppose we have a car that is moving from point A to B as shown in the figure below. Once it reaches point B, it starts moving again till point C. Now, to determine the net displacement of the car, we use the concept of vector addition. The net displacement of the car is given by the vector AC which can be calculated using the triangle law of vector addition as:
\(\overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC}\)
Similarly, if we have two vectors P and Q as given below and we need to find their sum, then we can move the vector Q in a way without changing its magnitude and direction such its tail is joined to the head of the vector P. Then, the sum of the vectors P and Q using the triangle law of vector addition is given by,
\(\overrightarrow{R} = \overrightarrow{P} + \overrightarrow{Q}\)
Triangle Law of Vector Addition Formula
Consider two vectors P and Q such that the angle between them is θ and their resultant sum vector using the triangle law of vector addition is given by the vector R. The formula for the magnitude R and direction ϕ of the resultant vector R using triangle law for the addition of vectors is given by,
R = √(P^{2} + Q^{2} + 2PQ cos θ)
ϕ = tan^{1}[(Q sin θ)/(P + Q cos θ)]
Triangle Law of Vector Addition Proof
Before getting on to the proof of the triangle law, let us first see the statement of the triangle law of vector addition:
Statement: If two vectors acting simultaneously on a body are represented both in magnitude and direction by two sides of a triangle taken in an order then the resultant sum vector (both magnitude and direction) of these two vectors is given by the third side of that triangle taken in the opposite order.
In the figure given below, consider two vectors P and Q whose magnitudes are given by the sides OA and AB, respectively. Now, the sum of these vectors using the triangle law of vector addition is given by the resultant vector R (side OB of the triangle) whose magnitude and direction are
 R = √(P^{2} + Q^{2} + 2PQ cos θ)
 ϕ = tan^{1}[(Q sin θ)/(P + Q cos θ)]
Now, extend the side OA till point C such that BC is perpendicular to OC and the angle between vectors P and Q is θ. Also, the direction of the resultant vector R is given by the angle ϕ. In rightangled triangle OBC, we have
OB^{2} = OC^{2} + BC^{2}
⇒ OB^{2} = (OA + AC)^{2} + BC^{2}  (1)
In the right triangle ABC, we have
cos θ = AC/AB and sin θ = BC/AB
⇒ AC = AB cos θ and BC = AB sin θ
⇒ AC = Q cos θ and BC = 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
Now, to find the direction of R, we have in right traingle OBC,
tan ϕ = BC/OC
⇒ tan ϕ = Q sin θ/(OA + AC) [From (2)]
⇒ tan ϕ = Q sin θ/(P + Q cos θ) [From (2)]
⇒ ϕ = tan^{1}[(Q sin θ)/(P + Q cos θ)] → Direction of the resultant vector R
Hence, we have proved the formulas for the triangle law of vector addition.
Important Notes on Triangle Law of Vector Addition
 Triangle law of vector addition is used to find the sum of two vectors when the head of the first vector is joined to the tail of the second vector.
 Magnitude of the resultant sum vector R: R = √(P^{2} + 2PQ cos θ + Q^{2})
 Direction of the resultant vector R: ϕ = tan^{1}[(Q sin θ)/(P + Q cos θ)]
Related Topics on Important Notes on Triangle Law of Vector Addition
Triangle Law of Vector Addition Examples

Example 1: Two vectors A and B have magnitudes of 4 units and 9 units and make an angle of 30° with each other. Find the magnitude and direction of the resultant sum vector using the triangle law of vector addition formula.
Solution: The formula for the resultant vector using the triangle law are:
R = √(A^{2} + B^{2} + 2AB cos θ)
ϕ = tan^{1}[(B sin θ)/(A + B cos θ)]
So, we have
R = √(A^{2} + B^{2} + 2AB cos θ)
= √(4^{2} + 9^{2} + 2 × 4 × 9 cos 30°)
= √(16 + 81 + 72 × √3/2)
= √(97 + 36√3)
= 12.623 units
The direction of R is given by,
ϕ = tan^{1}[(B sin θ)/(A + B cos θ)]
= tan^{1}[(9 sin 30°)/(4 + 9 cos 30°)]
= tan^{1}[(9 × 1/2)/(4 + 9 × √3/2)]
= tan^{1}[(4.5)/(11.8)]
= 20.87°
Answer: Hence, the magnitude of the resultant vector is 12.623 units and the direction is 20.87°, approximately.

Example 2: Two vectors with magnitudes 2 units and √2 units act on a body. The resultant vector has a magnitude of √10 units. Find the angle between the two given vectors.
Solution: Assume the two given vectors to be P and Q such that P = 2 and Q = √2.
Then, using the formula for the triangle law of vector addition, we have
R = √(P^{2} + 2PQ cos θ + Q^{2})
⇒ √10 = √(2^{2} + 2 × 2 × √2 cos θ + (√2)^{2})
⇒ 10 = 4 + 4√2 cos θ + 2
⇒ 4√2 cos θ = 10  4  2
⇒ cos θ = 4/4√2
⇒ cos θ = 1/√2
⇒ θ = 45°
Answer: Hence the angle between the two vectors is 45°.
FAQs on Triangle Law of Vector Addition
What is the Triangle Law of Vector Addition in Math?
Triangle Law of Vector Addition is a mathematical concept that is used to find the sum of two vectors. This law is used to add two vectors when the first vector's head is joined to the tail of the second vector and then joining the tail of the first vector to the head of the second vector to form a triangle, and hence obtain the resultant sum vector.
What is the Formula of Triangle Law of Vector Addition?
The sum of two vectors P and Q using the triangle law of vector addition is given by the resultant vector R whose magnitude and direction are:
 R = √(P^{2} + Q^{2} + 2PQ cos θ)
 ϕ = tan^{1}[(Q sin θ)/(P + Q cos θ)]
How Do You Add Vectors Using Triangle Law for the Addition of Vectors?
When two vectors P and Q are added, we add the vectors and determine the resultant vector R using the formulas:
 R = √(P^{2} + Q^{2} + 2PQ cos θ)
 ϕ = tan^{1}[(Q sin θ)/(P + Q cos θ)]
What is the Difference Between Parallelogram Law and Triangle Law of Vector Addition?
The resultant sum vector is given by the diagonal of a parallelogram using the parallelogram law and it is given by the third side of the triangle using the triangle law of vector addition.
Are Parallelogram Law and Triangle Law of Vector Addition the Same?
Parallelogram Law and Triangle Law of Vector Addition are equivalent methods of adding vectors. Both the laws give the same magnitude and direction of the resultant sum vector.
How To Prove the Triangle Law of Vector Addition Formula?
The triangle law of vector addition can be proved by extending the length of the side of the first vector and applying the formulas of the rightangled triangle.
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