**Triangle Law
of Vector Addition of Two Vectors**

The physical quantities may be broadly classified into the vectors and the scalars. The quantities with magnitude and direction both are known as vector quantities, it means a vector is a physical quantity that has both magnitude and direction. If two non-zero vectors are represented by the two sides of a triangle taken in same order then the resultant is given in opposite order.

i.e.., \(\overrightarrow{R}=\overrightarrow{A}+\overrightarrow{B}\) \(\because \overrightarrow{OB}=\overrightarrow{OA}+\overrightarrow{AB}\)

**1) Magnitude
of Resultant Vector: **In ΔABN, \(\cos \theta =\frac{AN}{B}\) ⇒ AN = B cos θ

\(\sin \theta =\frac{BN}{B}\) ⇒ BN = B sinθ; ΔOBN, we have OB² = ON² + BN².

R² = (A + B cosθ)² + (B sinθ)²

R² = A² + B² cos²θ + 2 AB cosθ + B² sin² θ

R² = A² + B² (cos²θ + sin²θ) + 2AB cosθ = A² + B² (1) + 2 AB cosθ

\(R=\sqrt{{{A}^{2}}+{{B}^{2}}+2AB\cos \theta }\).

**2) Direction of resultant vectors: **If θ is angle between \(\overrightarrow{A}\) and \(\overrightarrow{B}\), then

\(|\overrightarrow{A}+\overrightarrow{B}|\,=\,\sqrt{{{A}^{2}}+{{B}^{2}}+2AB\cos \theta }\) ; If \(\overrightarrow{R}\) makes an angle α with\(\overrightarrow{A}\), then in ΔOBN,

\(\tan \alpha =\frac{BN}{ON}=\frac{BN}{OA+AN}=\frac{B\sin \theta }{A+B\cos \theta }\).