Vectors

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**Scalar Quantities:** - Have only magnitude (size) - Examples: Mass (50 kg), Temperature (30°C), Speed (60 km/h) **Vector Quantities:** - Have both magnitude and direction - Examples: Displacement (

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**Unit Vector Definition:** A vector with magnitude 1 is called a unit vector, denoted as â. **Formula:** â = a⃗/|a⃗| **Step-by-Step Solution:** 1. Find |a⃗| = √(3² + 4²) = √(9 + 16) = √25 = 5 2. Un

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**Equal Vectors:** - Same magnitude: |a⃗| = |b⃗| - Same direction - Example: AB⃗ = CD⃗ if |AB⃗| = |CD⃗| and parallel **Collinear Vectors:** - Parallel to same line - a⃗ = kb⃗ (where k is scalar) - Ex

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**Triangle Law of Addition:** Place vectors head-to-tail, resultant joins tail of first to head of second. **Step-by-Step Solution:** 1. **Resultant:** R⃗ = a⃗ + b⃗ = (3î + 2ĵ) + (î - 4ĵ) 2. R⃗ = (3+

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**Direction Cosines:** l = cos α, m = cos β, n = cos γ **Proof:** Let vector r⃗ = xî + yĵ + zk̂ with magnitude |r⃗| = r **Step-by-Step:** 1. From geometry: x = r cos α, y = r cos β, z = r cos γ 2. T

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**Section Formula (Internal Division):** If P divides AB in ratio m:n internally, then: r⃗ₚ = (n·a⃗ + m·b⃗)/(m + n) **Given:** - A has position vector a⃗ = 2î + 3ĵ - k̂ - B has position vector b⃗ = 4

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**Scalar Product Formula:** a⃗·b⃗ = a₁b₁ + a₂b₂ + a₃b₃ Also, a⃗·b⃗ = |a⃗||b⃗|cos θ **Step-by-Step Calculation:** 1. **Scalar Product:** a⃗·b⃗ = (2)(1) + (-3)(2) + (1)(-2) a⃗·b⃗ = 2 - 6 - 2 = -6

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**Vector Product Using Determinant:** a⃗ × b⃗ = |î ĵ k̂| |2 1 -1| |1 -2 3| **Step-by-Step:** 1. **Expand determinant:** a⃗ × b⃗ = î(1×3 - (-1)×(-2)) - ĵ(2×3 - (-1)×1) + k̂(2×