Vector analysis Lecture 06 Chapter 3

1. General Physics (PHYS101) Vector analysis Lecture 06 Chapter 3

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

3. 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

4. y x Adding vectors Graphically triangle method

5. 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

6. 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.

7. 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

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

9. 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

10. Scalar Multiplication of Vectors • Associative law • Distributive law

11. y x Vector Subtraction

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

13. y x Vector Subtraction

14. General Physics (PHYS101) Golibjon Berdiyorov Building 6, Room 148 Vector analysis Lecture 06 Chapter 3

15. Vector is the sum of its components: y • How do we find and ? • Vector is decomposed to vectors and . • is the projection of the vector along the x-axis • is the projection of the vector along the x-axis x 0 Vector Decomposition

16. y y • Both and vectors • The vector is expressed as x x 0 0 Unit vectors • The magnitude of the unit vectors equals 1:

17. y y y • The vector is expressed as x x x 0 0 0 Unit vectors

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

19. Adding and subtracting vectors Algebraic method

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

21. Dot products

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

24. 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

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

26. Cross products

27. 1. The angle between and the negative y-axis is? 3. If has the magnitude of 3 m and makes an angle 30o with the +x axis, then the vector is? 4. 5. A vector is defined as Find the magnitude of a vector if the resultant of and is in the y-axis and its magnitude is 5.2. Vector analysis 2. A vector in the xy plane has a magnitude of 25 m and an x component of +12 m and a positive y component. Find the vector? The angle it makes with the positive y axis is?

28. 1. What is the angle between and 2. If what is the angle between them? Dot products

29. Cross products