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Vector Decomposition: Introduction, Definition, and Applications

Learn about the decomposition of a vector into its component vectors and the properties and applications of the dot (scalar) product. Find out how to use dot product to find angles, magnitudes, and perpendicularity between vectors, and understand the non-uniqueness of vector decomposition.

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Vector Decomposition: Introduction, Definition, and Applications

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  1. 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

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

  3. Dot (Scalar) Product of Two Vectors: ES2501: Statics/Unit 4-3: Decomposition of a Force 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.

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

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

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

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