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Theoretical Mechanics.Statics

Practical Lesson № 1

Vector Algebra

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Pythagorean theoremIn 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:

2 2 2 с a b .

900

bс

a

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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:

2 2 2 2 a b с b c cos .

b

с

aα

2 2 2

2

b с a

cos .b c

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The theorem of sinesSides of the triangle a, b, c, proportional to the sines respectively opposite angles α,β,γ:

a b c

.sin sin sin

b

с

aα

γβ

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Finding the values of segments

A

BO-1

-1

1

1 х

y

α

AB

sin AB OA sin .OA

OB

cos OB OA cos .OA

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Signs of trigonometric functions

+O-1

-1

1

1 х

y

sin(α)

+

– – +O-1

-1

1

1 х

y

cos(α)

+

–

–

+O-1

-1

1

1 х

y

tg(α) and

ctg(α)

+–

–

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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

6

4

3

2 2

3 3

4 5

6 3

2

1

2

1

2

3

2

3

2

3

2

3

2

1

3

1

31

3

1

3

1

2

3

3

1

2

3

3

2

2

2

2

2

2

2

2

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The general formulas of trigonometry• The relation between the trigonometric functions of the

same angle:

• Double angle formulas:

• Degree reducing formulas:

2 2sin cos 1.

sin 2 2 sin cos .

2 2cos 2 cos sin .

2 1 cos 2sin .

2

2 1 cos 2

cos .2

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Reduction of angles

x

sin(x) cos(α) cos(α) sin(α) –sin(α) –cos(α) –cos(α) –sin(α)

cos(x) sin(α) –sin(α) –cos(α) –cos(α) –sin(α) sin(α) cos(α)

tg(x) ctg(α) –ctg(α) –tg(α) tg(α) ctg(α) –ctg(α) –tg(α)

ctg(x) tg(α) –tg(α) –ctg(α) ctg(α) tg(α) –tg(α) –ctg(α)

2

2

32

3

2

2

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Vector conceptVector is a mathematical object, characterized by magnitude, direction and point of application.

A

 B

AB

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Projection of the vector onto the axis

A

B

u

α

 

Fu

��������������

uF F cos .

F

A'B'

 

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Projection of the vector onto the axis

A

B

u

 

F

 

Fu

��������������

uF F cos .

α

A'B'

 

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Rectangular components of a vector (slide 1)

A

B

O х

y

α

 

F

 

Fx>0

Fy>0

xF F cos .

yF F sin .

 

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A

B

O х

y

α

 

F

 

Fx<0

Fy>0

Rectangular components of a vector (slide 2)

 

xF F cos .

yF F sin .

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A

B

O х

y α

 

F

 

Fx<0

Fy<0

Rectangular components of a vector (slide 3)

 

xF F cos .

yF F sin .

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B

O х

y

α

 

F

 

Fx>0

Fy<0

A

Rectangular components of a vector (slide 4)

 

xF F cos .

yF F sin .

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Addition of vectors applied at the same point (Variant 1 – the parallelogram rule)

α

F

F1

F2

F1

F2

F

α

2 2

1 2 1 22 ����������������������������������������������������������������������F F F F F cos .

1 2 ������������������������������������������F F F .

��������������x x y y z zF a b ;a b ;a b .

1

2

��������������

��������������x y z

x y z

F a ;a ;a ;

F b ;b ;b .

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Addition of vectors applied at the same point (Variant 2 – the triangle rule)

F1

F2

F1

F2

F1

F2

FO O O

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Addition of vectors applied at the same point (Variant 3 – the polygon rule)

F1

F2

F1

F2

F

O

F3

O

F3

F1

F2O

F3

Subtraction of vectors applied at the same point

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α

FF1

F2

1 2 ������������������������������������������F F F

F1

O O

-F2

2 2

1 2 1 22 ����������������������������������������������������������������������F F F F F cos .

1 2 ������������������������������������������F F F

1

2

x y z

x y z

F a ;a ;a ;

F b ;b ;b .

��������������

��������������

x x y y z zF a b ;a b ;a b . ��������������

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Resolution of a vector into components along the coordinate axes

O х

y

F1

F2

 

F

O х

y F1

F2

 F

A

A

O х

y F1

F2F

O х

y

F1

F2F

A

A

 

 

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m

l

Fm

Fl F

A

Resolving of a vector into components

along arbitrary directions

23

Remember!Notation:

- vector;

F or - vector magnitude;

Fx, Fm - component of the vector along the axis

(scalar value);

- vector component along the direction (vector value).

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F

Fx

F

Dot product of two vectors

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α a

bα

ab

a b a b cos .

25

Dot product of the unit vectors of Cartesian rectangular coordinate system

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z

y

x

O

0 1

0 1

0 1

i i i i cos ,

j j j j cos ,

k k k k cos .

90 0

90 0

90 0

i j i j cos ,

j k j k cos ,

k i k i cos .

1i j k

i

j

k

Dot product of the vector and the unit vector of axis (slide 1)

Projection of the vector onto the axis is equal to the dot product of the vector and the unit vector of axis21/04/2023 26

xi

F

α

����������������������������������������������������������������������

xF i F i сos F сos F .

Fx

х

y

z

 

 

O

   

F

Fx α

 

2D 3D

xFcosF

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х

y

z

 

 

O

   

 

FFy

х

y

z

 

 

O

   

 

F

Fz

βγ

y

F j F j cos

F cos F

z

F k F k cos

F cos F

Dot product of the vector and the unit vector of axis (slide 2)

yFcosF

zFcosF

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Dot product of vectors in coordinate form

x y z x y z

x x y y z z

a b a i a j a k b i b j b k

a b a b a b .

2 2 2

2

0

0

x x y y z z x y za a a a cos a a a a a a a a a ,

a a a a cos a ,

2 2 2x y za a a a .

!!!

Vector magnitude

x x y y z za b a b a ba b

cos .a b a b

Cosine of an angle between two vectors

O

Relation between the directions cosines of vector (3D)

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2 2 2 2x y za a a a ,

2 2 21 cos cos cos

х

y

z

 

 

   

 

FFy

Fz

βγ

Fxα

22 2

2 2 2

2 2 2

1 yx zaa a

a a a

cos cos cos .

2 2 2 2

2 2

x y za a aa,

a a

29

Cross product of two vectors

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.

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αa

d

d a b

b

The result of cross product – vector.Magnitude of the cross product:

d a b a b Sin

z

x

Oy

d a b a b sin .

31

d a b,

a,b yOz, d Ox

Examples of determining the direction of the cross product of two vectors

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α

a

d

b

z

x

O

d a b

α

a

d

b

z

xO

yy

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Example of determining the direction of the cross product of two vectors

33

Cross product of the unit vectors of Cartesian rectangular coordinate system

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z

y

x

O

0 0

0 0

0 0

i i i i sin ,

j j j j sin ,

k k k k sin .

i j k ,

j k i ,

k i j .

1i j k

i

j

k

j i k ,

k j i ,

i k j .

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

determinant

y z x yx zx y z

x zy z x yx y z

i j ka a a aa a

d a a a i j k .b bb b b b

b b b

y zx y z z y

y z

a ad a b a b

b b x z

у z x x zx z

a ad a b a b

b b

x yz x y y x

x y

a ad a b a b

b b

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Summary slide – Information about all topics studied during the lesson.

It is compiled by student him/herself!