Motion in a Plane

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**Scalar Quantity**: Has only magnitude, no direction. Examples: mass (50 g), density (1000 kg/m³), temperature (25°C), speed (10 m/s). **Vector Quantity**: Has both magnitude and direction. Examples

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**Triangle Law of Vector Addition**: If two vectors are represented in magnitude and direction by the two sides of a triangle taken in order, then their resultant is represented by the third side of t

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For two vectors **A** and **B** with angle θ between them: **Derivation**: Using parallelogram law and dropping perpendicular RT: (PR)² = (PT)² + (RT)² = (PQ + QT)² + (RT)² = A² + B²cos²θ + 2AB cosθ

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**Special Cases**: 1. **Parallel vectors** (θ = 0°): R = A + B Vectors in same direction, maximum resultant 2. **Perpendicular vectors** (θ = 90°): R = √(A² + B²) cosθ = 0, Pythagorean theorem

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**Scalar Product**: The scalar product of two vectors **A** and **B** is a scalar quantity equal to the product of their magnitudes and cosine of angle between them. **Formula**: **A** · **B** = AB c

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**Vector Product**: **A** × **B** = C, where |C| = AB sinθ and direction is perpendicular to plane containing **A** and **B**. **Right-Hand Rule**: Curl fingers of right hand from **A** to **B** thro

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**Dot Product Relations**: î · î = ĵ · ĵ = k̂ · k̂ = 1 î · ĵ = ĵ · k̂ = k̂ · î = 0 **Cross Product Relations**: î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ ĵ × î = -k̂, k̂ × ĵ = -î, î × k̂ = -ĵ î × î = ĵ × ĵ

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**Given**: **A** = 4î + 3ĵ **Magnitude**: |**A**| = √(Ax² + Ay²) = √(4² + 3²) = √(16 + 9) = √25 = 5 units **Direction**: tanθ = Ay/Ax = 3/4 = 0.75 θ = tan⁻¹(0.75) = 36.87° with x-axis **Answer**: M