ES2501: Statics/Unit 4-1: Decomposition of a Force

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ES2501: Statics/Unit 4-1: Decomposition of a Force Introduction: Decomposition ---- express a vector as a sum of its component vectors, which is an inverse operation of summation. Decomposition of a vector is non-unique unless the direction of decomposition is fully specified. A vector can be expressed as sum of its parallel and perpendicular components, as a special case; Dot (scalar) product of two vectors is a useful tool for vector decomposition

ES2501: Statics/Unit 4-2: Decomposition of a Force Dot (Scalar) Product of Two Vectors: Definition result is a scalar Properties

ES2501: Statics/Unit 4-3: Decomposition of a Force Dot (Scalar) Product of Two Vectors: unit vectors Application Use dot product to find the angle between two vectors Use dot product to find magnitude of a vector Use dot product to find if two vectors are perpendicular Dot product in terms of Cartesian components: Example: Given Find the angle between these two vectors.

ES2501: Statics/Unit 4-4: Decomposition of a Force Given directions --- unit directional vectors perpendicular component of Projection of on or parallel component of

ES2501: Statics/Unit 4-5: Decomposition of a Force Non-Uniqueness of Decomposition Note: decomposition is not unique unless directions are given

ES2501: Statics/Unit 4-6: Decomposition of a Force Example: Find the components of parallel and perpendicular to member AB. Note that