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

3. 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.

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

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

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