Theoretical Mechanics . Statics

1. Theoretical Mechanics.Statics Vector Algebra Practical Lesson № 1

2. Pythagoreantheorem In a right triangle, the square of the hypotenuse length cis equal to the sum of the squares of the side aand b lengths: с b 900 a

3. The theorem of cosines For a flat triangle with sides a, b, c and the angle α, which is the opposite to side a, the square of the triangle side is equal to the sum of the squares of the other two sides minus doubled product of these sides and the cosine of the angle between them: b a α с

4. The theorem of sines Sides of the triangle a, b, c, proportional to the sinesrespectively opposite angles α,β,γ: b a γ α β с

5. y Finding the values of segments 1 A α х -1 O B 1 -1

6. Signs of trigonometric functions y y 1 1 – + + + х х -1 -1 O O 1 1 – – – + y -1 -1 sin(α) cos(α) 1 – + х -1 O 1 + – tg(α) and ctg(α) -1

7. Values ​​of trigonometric functions

8. The general formulas of trigonometry • The relation between the trigonometric functions of the same angle: • Double angle formulas: • Degree reducing formulas:

9. Reduction of angles

10. Vector concept Vector is a mathematical object, characterized by magnitude, direction and point of application. B AB A

11. Projection of the vector onto the axis B A u α B' Fu A'

12. Projection of the vector onto the axis B F α A u A' Fu B'

13. Rectangular components of a vector (slide 1) y B F Fy>0 α A х O Fx>0

14. Rectangular components of a vector (slide 2) y B F Fy>0 α A х O Fx<0

15. Rectangular components of a vector (slide 3) α y A F Fy<0 B х O Fx<0

16. Rectangular components of a vector (slide 4) y A F α Fy<0 B х O Fx>0

17. Addition of vectors applied at the same point (Variant 1 – the parallelogram rule) F F2 F2 F α α F1 F1

18. Addition of vectors applied at the same point (Variant 2 – the triangle rule) F2 F2 F2 F O O O F1 F1 F1

19. Addition of vectors applied at the same point (Variant 3 – the polygon rule) F2 F2 F F2 O O O F1 F1 F3 F3 F1 F3

20. Subtraction of vectors applied at the same point -F2 F F2 α F1 F1 O O

21. Resolution of a vector into components along the coordinate axes A y y A O O F F2 F F2 F2 F F2 F х х F1 A y y A F1 F1 F1 O O х х

22. Resolving of a vector into components along arbitrary directions l Fl F Fm m A

23. Remember!Notation: F F -vector; For- vector magnitude; Fx, Fm-component of the vector along the axis (scalar value); -vector component along the direction (vector value). Fx

24. Dot product of two vectors b b α α a a

25. z Dot product of the unit vectors of Cartesian rectangular coordinate system y O x i j k

26. 3D 2D Dot product of the vector and the unit vector of axis (slide 1) z F F Projection of the vector onto the axis is equal to the dot product of the vector and the unit vector of axis O α Fx α y x Fx х i

27. Dot product of the vector and the unit vector of axis (slide 2) z z Fz γ F F Fy O β O y y х х

28. Vector magnitude Dot product of vectors in coordinate form !!! Cosine of an angle between two vectors

29. z Relation between the directions cosines of vector (3D) Fz γ F Fy β O Fx y α х

30. The result of cross product – vector. Magnitude of the cross product: Cross product of two vectors z d b α The result of cross product is perpendicular to the plane in which the multiplied vectors a and b are situated; in this case the vectors a, b and d form a right-hand triad. O a y x

31. Examples of determining the direction of the cross product of two vectors z α z α b b d d O y O y x x a a

32. Exampleof determining the direction of the cross product of two vectors

33. z Cross product of the unit vectors of Cartesian rectangular coordinate system y O x i j k

34. Calculation of the cross product of vectors assigned in Cartesian rectangular coordinate system by the formal determinant

35. Summary slide – Information about all topics studied during the lesson. It is compiled by student him/herself!