Triangle Law of Vector Addition

Triangle Law of Vector Addition is one of the laws, which help us add two vectors and as well as visualize them. Vectors are the backbone of many technologies nowadays, such as computer graphics, visual effects, machine learning, and artificial intelligence. Therefore, understanding the addition of vectors is a much-needed skill to understand these further advanced topics.

In this article, we will help you understand the basic concept of vector addition and how the Triangle Law of Vector Addition is used to explain the concept of vector addition. Other than that, we will also learn about the formula for the resultant vector and its derivation, and also learn to solve various problems related to the Triangle Law of Vector Addition.

What is Vector Addition?

Vector quantity is the quantity which contains both magnitude and direction and the procedure of adding two or more vectors is called vector addition. The addition of two vectors is different from traditional algebraic additions as in the case of vectors we need to add their magnitude as well as their direction i.e., the magnitude and direction of the resultant vector depends on the added vectors.

For the addition of two vectors, there are some necessary conditions that have to be followed. First, to perform the addition we require two vector quantities only. The quantities of different forms i.e., scalar and vector cannot be added. Also, the vector added must be of the same type as different types of vectors cannot be added together.

Since vector addition is not similar to regular algebraic additions, we require some specific laws to perform the addition of vectors. The following are two laws for vector addition:

• Triangle Law of Vector Addition
• Parallelogram Law of Vector Addition
• Polygon Law of Vector Addition

Learn more about Vector Addition

What is Triangle Law of Vector Addition?

The triangle law of vector addition states that when the two vectors are represented by the two sides of the triangle, then the third side of the triangle represents the resultant vector of addition i.e., the third side of the triangle represents both the magnitude and direction (opposite to the direction of given vectors) of the resultant vector.

If and are two vectors. We have to add these two vectors, then the resultant vector according to triangle law of vector addition is given by:

Which can be illustrated using the following diagram.

From the two given vectors, to form triangle we arrange these two vectors in such a way that the tail of one vector is joined to the head of the other vector.

Triangle Law of Vector Addition Formula

The triangle law of vector addition arranges the two vector and its resultant vector of addition in the form of a triangle. In this triangle we have the third side of the triangle as resultant vector R and an angle θ between two vectors.

Formula for Magnitude of Resultant of any two vectors is given by

|R| = √(A²+ B² + 2ABcosθ)

where,

• R is the Resultant of A and B
• A and B are two vectors
• θ is the angle between A and B

Formula for the direction of resultant vector of A and B i.e., Φ; is given by:

Φ = tan^-1[Bsinθ /(A + Bcosθ)]

where,

• Φ is the angle of the Vector from positive x-axis
• A and B are two vectors
• θ is the angle between A and B

Triangle Law of Vector Addition Derivation

Consider two vectors A and B representing the two sides of the triangle OP and PQ respectively. Let vector R (OQ) be the resultant vector of the addition of A and B.

According to the above description, we draw the below diagram.

From the triangle OSQ,

OQ² = OS² + QS²

OQ² = (OP +PS)² + QS² ——(1)

In triangle PSQ with θ as the angle between A and B

cos θ = PS / PQ

PS = PQ cosθ = B cosθ

sin θ = QS / PQ

QS = PQ sinθ = B sinθ

Substituting the values of PS and QS in equation (1), we get

R² = (A + Bcosθ)² + (Bsinθ)²

R² = A² + 2ABcosθ + B²cos²θ + B²sin²θ

R² = A² + 2ABcosθ + B²

Therefore,

R = √(A²+ B² + 2ABcosθ)

The above equation represents the magnitude of resultant vector.

To find the direction of the resultant vector R, let Φ be the angle between vectors A and R.

From triangle, OQS

tanΦ = QS / OS

tanΦ = QS / (OP + PS)

tanΦ = Bsinθ / (A + Bcosθ)

therefore,

Φ = tan^-1[Bsinθ / (A + Bcosθ)]

The above equation gives the direction of the resultant vector R.

Read More,

• Vector Algebra
• Vector Operations
• Vector Calculus

Practice Problems on Triangle Law of Vector Addition

Problem 1: Car travelling 40 km West and 30 km South. Calculate the resultant displacement using Triangle Law of Vector Addition.

Problem 2: A man walks 8 km at an angle of 60 degrees South of West and then 8 km West. Determine the resultant displacement.

Problem 3: An airplane travelling 800 km North and then 500 km due West. Find the resultant of displacement of airplane.

Problem 4: Boat travelling at 10 km upstream and 16 km downstream find the total displacement of boat.

Solved Examples on Triangle Law of Vector Addition

Example 1: Two vectors P and Q have magnitudes of 9 units and 16 units and make an angle of 30° with each other. Using triangle law of vector addition, find the magnitude and direction of resultant vector.

Solution:

According to the triangle law of vector addition

|R| = √(A²+ B² + 2ABcosθ)

Φ = tan^-1[Bsinθ /(A + Bcosθ)]

The magnitude of R :

|R| = √[9² + 16² + 2(9)(16)cos 30]

|R| = √[81 + 256 + 288(√3 / 2)]

|R| = √[337 + 144√3]

|R| = 24.21 units

The direction of R:

Φ = tan^-1[16sin30 /(9 + 16cos30)]

Φ = tan^-1[16(1/2) /(9 + 16(√3/2))]

Φ = tan^-1[8 /(9 + 8√3)]

Φ = 19.29°

Example 2: Two vectors have magnitudes 3 and √3 units. The resultant vector has the magnitude √21 units. Find the angle between the two vectors.

Solution:

According to the triangle law of vector addition

R² = (A²+ B² + 2ABcosθ)

(√21)² = [3²+ (√3)² + 2(3) (√3)cosθ]

21 = [9 + 3 + 2(3) (√3)cosθ]

21 = [12 + 6√3cosθ]

21 – 12 = 6√3cosθ

9 = 6√3cosθ

cosθ = √3 / 2

θ = cos^-1(√3 / 2)

θ = 30°

The angle between two vectors is θ = 30°.

Example 3: Consider two vectors A and B where, . Find the resultant vector after the addition of two vectors.

Solution:

According to triangle law of vector addition

The resultant vector is

Example 4: Find the magnitude of the vector P, given that magnitude of vector Q and resultant vector R is 4 and 6 units respectively. The angle between two vectors is 60°.

Solution:

According to triangle law of vector addition, the magnitude of R is given by:

R² = [P² + Q² + 2PQcosθ]

Here, R = 6, Q = 4 and θ = 60°

Putting above values in the formula to find the magnitude of vector P.

6² = P + 4² + 2P(4)cos 60]

36 = P + 16 + (8P/ 2)

36 = P + 16 + 4P

5P = 36 – 16

5P = 20

P = 4 units

The magnitude of vector P = 4 units.

Example 5: Find the magnitude of vector A, if the magnitude of vector B is 10 units, angle between two vectors is 60° and the angle between vector A and the resultant vector is 45°.

Solution:

According to the triangle law of vector addition

Φ = tan^-1[Bsinθ /(A + Bcosθ)]

tanΦ = [Bsinθ /(A + Bcosθ)]

Here, B = 10 units, θ = 60° and Φ = 45°

Putting these values in the above formula to obtain the magnitude of vector A.

tan 45 = [10sin60 /(A + 10cos60)]

1 = [10(√3 / 2)] / [A + 10(1/2)]

1 = 5√3 / [A + 5]

A + 5 = 5√3

A = 5√3 – 5

A = 5(√3 – 1)

A = 3.66 units

The magnitude of vector A is 3.66 units.

FAQs on Triangle Law of Vector Addition

1. What are Scalar and Vector Quantities?

The quantities which have only magnitude are called scalar quantities. The quantities which have both magnitude and direction are called vector quantities.

2. What is Vector Addition?

The method of addition of vector using laws is called as vector addition.

3. What are the Necessary Conditions for Vector Addition?

The necessary conditions for vector addition are:

• All the quantities added must be a vector quantity.
• The vectors which are added must be of same type.

4. What are the Three laws of Vector Addition?

The three laws of vector addition are:

• Triangle Law of Vector Addition
• Parallelogram Law of Vector Addition
• Polygon Law of Vector Addition

5. State Triangle law of Vector Addition.

The triangle law of vector addition states that when two vectors are added, it can be represented by the two sides of the triangle and the resultant vector is given by the third side.

6. Write the formula for the triangle law of vector addition.

The formula for the triangle law of vector addition:

|R| = √(A²+ B² + 2ABcosθ)

Φ = tan^-1[Bsinθ /(A + Bcosθ)]