Vector Algebra

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A vector is a quantity that has both magnitude and direction. Examples: displacement, velocity, force. Scalar is a quantity with only magnitude (no direction). Examples: mass, time, temperature, spee

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Position vector of a point P(x,y,z) with respect to origin O is the vector OP. Formula: r⃗ = OP⃗ = xî + yĵ + zk̂ Magnitude: |r⃗| = √(x² + y² + z²) The coordinates (x,y,z) are called scalar componen

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Direction cosines: If a vector makes angles α, β, γ with positive x, y, z axes respectively, then cos α, cos β, cos γ are direction cosines (denoted as l, m, n). Property: l² + m² + n² = 1 Direction

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Zero vector (0⃗): Initial and terminal points coincide, magnitude = 0 Unit vector (â): Magnitude = 1, â = a⃗/|a⃗| Equal vectors: Same magnitude and direction regardless of position Collinear vector

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Triangle Law: If two vectors are represented by two sides of a triangle taken in order, then their sum is represented by the third side taken in opposite order. Formula: AB⃗ + BC⃗ = AC⃗ Note: When s

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Parallelogram Law: If two vectors are represented by two adjacent sides of a parallelogram, then their sum is represented by the diagonal of the parallelogram passing through their common point. Both

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1. Commutative: a⃗ + b⃗ = b⃗ + a⃗ 2. Associative: (a⃗ + b⃗) + c⃗ = a⃗ + (b⃗ + c⃗) 3. Additive Identity: a⃗ + 0⃗ = a⃗ 4. Additive Inverse: a⃗ + (-a⃗) = 0⃗ These properties make vector addition simi

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If a⃗ = a₁î + a₂ĵ + a₃k̂ and b⃗ = b₁î + b₂ĵ + b₃k̂ Addition: a⃗ + b⃗ = (a₁+b₁)î + (a₂+b₂)ĵ + (a₃+b₃)k̂ Subtraction: a⃗ - b⃗ = (a₁-b₁)î + (a₂-b₂)ĵ + (a₃-b₃)k̂ Scalar multiplication: λa⃗ = λa₁î + λa₂