Theoretical Mechanics Statics Practical Lesson 1 Vector Algebra

Theoretical Mechanics. Statics Practical Lesson № 1 Vector Algebra 26/01/2022 1

Pythagorean theorem In a right triangle, the square of the hypotenuse length c is equal to the sum of the squares of the side a and b lengths: с b 26/01/2022 900 a 2

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 α a с 26/01/2022 3

The theorem of sines Sides of the triangle a, b, c, proportional to the sines respectively opposite angles α, β, γ: b α γ β a с 26/01/2022 4

Finding the values of segments y A α O 26/01/2022 B х 5

Signs of trigonometric functions y 1 – + + -1 sin(α) O – – 1 х -1 y 1 -1 – -1 tg(α) and 26/01/2022 ctg(α) O – cos(α) + + х 1 -1 + O + – 1 х -1 6

Values of trigonometric functions 0 00 300 450 600 900 1200 1350 1500 1800 2700 sin(α) 0 1 0 – 1 cos(α) 1 0 – 1 0 tg(α) 0 1 – – 1 0 – ctg(α) – 1 0 – 1 – 0 26/01/2022 7

The general formulas of trigonometry • The relation between the trigonometric functions of the same angle: • Double angle formulas: • Degree reducing formulas: 26/01/2022 8

Reduction of angles x sin(x) cos(α) cos(x) sin(α) –sin(α) tg(x) ctg(α) ctg(x) tg(α) 26/01/2022 sin(α) –cos(α) –sin(α) cos(α) –ctg(α) –tg(α) –ctg(α) –ctg(α) 9

Vector concept Vector is a mathematical object, characterized by magnitude, direction and point of application. AB B A 26/01/2022 10

Projection of the vector onto the axis B A u α A' 26/01/2022 B' Fu 11

Projection of the vector onto the axis B F F B' u 26/01/2022 A α u A' 12

Rectangular components of a vector (slide 1) y B Fy>0 O 26/01/2022 A F α Fx>0 х 13

Rectangular components of a vector (slide 2) y B Fy>0 F α A O 26/01/2022 Fx<0 х 14

Rectangular components of a vector (slide 3) α y F Fy<0 A B O 26/01/2022 Fx<0 х 15

Rectangular components of a vector (slide 4) y A α Fy<0 F B O 26/01/2022 Fx>0 х 16

Addition of vectors applied at the same point (Variant 1 – the parallelogram rule) F F 2 α F 1 26/01/2022 F α F 2 F 1 17

Addition of vectors applied at the same point (Variant 2 – the triangle rule) F 2 F 1 26/01/2022 F 1 18

Addition of vectors applied at one point (Variant 3 – the polygon rule) F 2 O F 1 F 3 26/01/2022 O F 1 F 3 F 2 F 3 F 19

Subtraction of vectors applied at the same point F 1 F -F 2 O 26/01/2022 F 1 α O 20

Resolution of a vector into components along the coordinate axes y y F 2 F A A F 1 F F 2 F 1 O F 1 y F х O A y х F F 2 F 1 O 26/01/2022 х O F 2 A х 21

Resolving of a vector into components along arbitrary directions l Fl A 26/01/2022 F m Fm 22

F - vector; Remember! Notation: F or F - vector magnitude; Fx, Fm - component axis (scalar value); of the vector along the Fx - vector component along the direction (vector value). 26/01/2022 23

Dot product of two vectors b b α a 26/01/2022 α a 24

Dot product of the unit vectors of Cartesian rectangular coordinate system z k O j y i x 26/01/2022 25

Dot product of the vector and the unit vector of axis (slide 1) 2 D z F O α i 26/01/2022 Fx 3 D Fx F α x y х Projection of the vector onto the axis is equal to the scalar product of the vector and the unit vector of axis 26

Dot product of the vector and the unit vector of axis (slide 2) z O z Fz F β Fy O γ F y y х х 26/01/2022 27

Dot product of vectors in coordinate form Vector magnitude !!! Cosine of an angle between two vectors 26/01/2022 28

Relation between the directions cosines of vector (3 D) z Fz Fx F γ β Fy O α y х 26/01/2022 29

Vector product of two vectors The result of vector product – vector value. Magnitude of the vector product: z d b α O a y x 26/01/2022 30

Examples of determining the direction of the vector product of two vectors a z α O x 26/01/2022 d b b z y O x α a y d 31

Vector product of the unit vectors of Cartesian rectangular coordinate system z k O j y i x 26/01/2022 32

Calculation of the vector product of vectors assigned in Cartesian rectangular coordinate system by the formal determinant 26/01/2022 33

Summary slide – Information about all topics studied during the lesson. It is compiled by student him/herrself! 26/01/2022 34