1. Forces Classification of Forces Force System
1 – Concurrent : all forces pass through a point
2 – Coplanar : in the same plane
3 – Parallel : parallel line of action
4 – Collinear : common line of action
Three Types
1 – Free (direction, magnitude and sense)
2 – Sliding
3 – Fixed A O
Origin

2. Trigonometric Relations of a triangle
Phythagorean theorem is valid only for a right angled triangle. For any triangle (not necessarily right angled)
Sine Law
Cosine Law A B C

3. Cartesian Coordinate Systems
A much more logical way to add/subtract/manipulate vectors is to represent the vector in Cartesian coordinate system. Here we need to find the components of the vector in x,y and z directions.
Simplification of Vector Analysis x y A z n en i j k x y z

4. Vector Representation in terms unit vector
Suppose we know the magnitude of a vector in any
arbitrary orientation, how do we represent the vector?
Unit Vector : a vector with a unit magnitude A en ^

5. Right-Handed Coordinate Systemx y z
Right-Handed System
If the thumb of the right hand points in the direction of the positive z-axis when the fingers are pointed in the x-direction & curled from the x-axis to the y-axis.

6. Components of a Vector y A Ay Ax Az z x In 3-D y A Ay Ax x In 2-D
Cartesian (Rectangular) Components of a Vector y A Ay Ax Az z x
In 3-D y A Ay Ax x
In 2-D

7. Cartesian Vectors z Az = Azk A = Aen en y Ax = Axi Ay = Ayj x
Cartesian Unit Vectors y
A = Aen
Ay = Ayj
Ax = Axi
Az = Azk z x n i j k en

8. Cartesian Vectors
Magnitude of a Cartesian Vector y A Ay Ax Az z x

9. Addition and Subtraction of vectorsy F Fy Fx Fz z x

10. Force Analysis y F Fy Fx Fz z x

11. Force Analysis
Ex: y z x
10ft
6ft
8ft
F = 600

12. Summary of the Force Analysisz F y F x q =
cos - 1 y y F

13. Addition/Subtraction of Cartesian vectors
Since any vector in 3-D can be expressed as components in x,y,z directions, we just need to add the corresponding components since the components are scalars.
Then the addition
Then the subtraction

14. ΑΣΚΗΣΗ-36
Solution:

15. ΑΣΚΗΣΗ-36 (2) Unit vector and direction cosines
From these components, we can
determine the angles

16. Position Vectors z
(xb, yb, zb) r
( xa, ya, za) y x
Position vectors can be determined using the coordinates of the end and beginning of the vector

17. Force Vector Along a Line
A force may be represented by a magnitude & a position
Force is oriented along the vector AB (line AB) B z F
Unit vector along
the line AB A y x

18. Dot Product y x A
Dot Product (Scalar Product)

19. Application of dot product
Angle between two vectors A B