1. Outline Addition and subtraction of vectors Vector decomposition
Unit vectors
Dot (scalar) product of vectors
Cross (vector) product of vectors

2. Adding vectors
When adding vectors, their directions must be taken into account.
Units must be the same.
Graphical methods
Use scale drawings
Algebraic methods
More convenient

3. Adding vectors Graphically
triangle method y x

4. Adding vectors Graphically
Continue drawing the vectors “tip-to-tail”
The resultant vector is drawn from the origin of the first vector to the and of the second one.
Measure the length of the resultant vector and its angle

5. Adding vectors Graphically
When you have many vectors, just keep repeating the process until all are included
The resultant is still drawn from the origin of the first vector to the end of the last vector.

6. Alternative Graphical Method
When you have only two vectors, you may use the Parallelogram Method
All vectors, including the resultant, are drawn from a common origin

7. Properties of Vector addition
Vectors obey the Commutative Law of Addition
The order in which the vectors are added does not affect the result

8. Properties of Vector addition
Vectors also obey the Associativity Law of Addition
When adding three vectors, it does not matter which two yo start with

9. Scalar Multiplication of Vectors
Associative law
Distributive law

10. Vector Subtraction y x

11. Vector Subtraction Special case of vector addition
If A-B, then use A+B:
Continue with standard vector addition procedure

12. Vector Subtraction y x
𝑫=𝑨−𝑩

13. Vector Decomposition Vector is decomposed to vectors and .
is the projection of the vector along the x-axis y x
is the projection of the vector along the x-axis
Vector is decomposed to vectors and .
Vector is the sum of its components:
How do we find and ?

14. Unit vectors y Both and vectors x
Both and vectors
The magnitude of the unit vectors equals 1: y x
The vector is expressed as

15. Unit vectors y x
The vector is expressed as y x y x

16. Unit vector in 3D cartesian coordinates
Unit vector in the directions of vector

17. Adding and subtracting vectors
Algebraic method

18. Dot product of vectors
Dot product (or scalar product) of vectors and is defined as
Dot product is always a scalar quantity
Two vectors are orthogonal (i.e. perpendicular to each other) if their dot product is zero

19. Dot products

20. Cross product of vectors
Cross product is a vector operation that generates a new vector from the other two vectors.
The magnitude of cross product of vectors and is defined as
Cross product is always a vector perpendicular to the plane.

22. Properties of cross product
The cross product is anti-commutative since changing the order of the vectors cross product changes the direction of the resulting vector

23. Mathematical definition of cross product
Two vectors are parallel to each other if and only if:

24. Cross products