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, [Tex]\overrightarrow{\rm A}= 3\hat{i} + 5\hat{j}, \overrightarrow{\rm B}= 6\hat{i} – 2\hat{j} [/Tex]. Find the resultant vector [Tex]\overrightarrow{\rm R} [/Tex] after the addition of two vectors.

Solution:

[Tex]\overrightarrow{\rm A}= 3\hat{i} + 5\hat{j}, \overrightarrow{\rm B}= 6\hat{i} – 2\hat{j} [/Tex]

According to triangle law of vector addition

[Tex]\overrightarrow{\rm R}=\overrightarrow{\rm A}+\overrightarrow{\rm B}\\ \overrightarrow{\rm R} = (3\hat{i} + 5\hat{j}) +(6\hat{i}-2\hat{j})\\ \overrightarrow{\rm R} = (9\hat{i} + 3\hat{j}) [/Tex]

The resultant vector is [Tex]\overrightarrow{\rm R} = (9\hat{i} + 3\hat{j}) [/Tex]

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.

The Triangle Law of Vector Addition is a method used to add two vectors. It states that when two vectors are represented as two sides of a triangle in sequence, the third side of the triangle is taken in the opposite direction. It represents the resultant vector in both magnitude and direction.

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.

Let’s learn more about Triangle Law of Vector Addition in detail with steps to add two vectors with formula below.