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## Triangle's Law Of Vector Addition The addition of two vectors can be represented by a triangle: - The two vectors are represented by two sides of the triangle, taken in the same order. - The resultant vector of the two vectors is represented by the third side of the triangle, taken in opposite...

## Triangle's Law Of Vector Addition The addition of two vectors can be represented by a triangle: - The two vectors are represented by two sides of the triangle, taken in the same order. - The resultant vector of the two vectors is represented by the third side of the triangle, taken in opposite order. ### Graphical Method: 1. Represent vector **A** by line **OP**. 2. At point **P**, draw line **PQ** representing vector **B**. 3. Join **OQ**. This line represents the resultant **R** of **A** and **B**. ### Analytical Method: 1. Consider two vectors **A** and **B** acting at a point. 2. Represent these vectors by sides **OP** and **PQ** of triangle **OPQ**. 3. The third side **OQ** of the triangle represents the resultant **R**. - In right triangle **OTQ**: - **OQ** = **OP** + **PT** - **QT** = **B sin θ** - **PT** = **B cos θ** Therefore, the resultant vector **R** can be calculated as follows: - **R2** = **OQ2** + **QT2** = (**OP** + **PT**)2 + **QT2** - **R2** = **OP2** + **PT2** + 2 **OP x PT** + **QT2** - **R2** = **A2** + **B2 cos2 θ + 2 **AB cos θ** + **B2 sin2 θ** - **R2** = **A2** + **B2** (**cos2 θ + sin2 θ**) + 2 **AB cos θ** - **R2** = **A2** + **B2** + 2 **AB cos θ** - **R** = √(**A2** + **B2** + 2 **AB cos θ**) The direction of the resultant **R** (angle **β**) can be found as: - **tan β** = **QT / OT** = **B sin θ / (A + B cos θ)** - **β** = **tan-1 (B sin θ / (A + B cos θ))** **Therefore, the magnitude and direction of the resultant vector are determined.**