Experimental Physics - Mechanics - Vectors and Scalars 1

Experimental Physics

EP1 MECHANICS

- Vectors and Scalars -

Rustem Valiullin

https://bloch.physgeo.uni-leipzig.de/amr/

Experimental Physics - Mechanics - Vectors and Scalars 2

Scalar and vector quantities

A scalar quantity is specified by a single value with an appropriate unit and has no direction.

A vector quantity has both magnitude and direction.

Experimental Physics - Mechanics - Vectors and Scalars 3

A

B

Displacement vector

r

𝑟

Experimental Physics - Mechanics - Vectors and Scalars 4

Adding vectors

𝐵

𝐴

𝐶

Commutative law: 𝑎 + 𝑏 = 𝑏 + 𝑎

Associative law: 𝑎 + (𝑏 + 𝑐 ) = (𝑎 + 𝑏) + 𝑐

Vector subtraction: 𝑎 − 𝑏 = 𝑎 + (−𝑏)

𝐶 = 𝐴 + 𝐵

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Thumb

Index finger

Middle finger

Ring finger

Little finger

• A Cartesian coordinate system consists of

three mutually perpendicular axes, the x-, y-,

and z-axes.

• By convention, the orientation of these axes is

such that when the index finger , the middle

finger, and the thumb of the right-hand are

configured so as to be mutually perpendicular.

• The index finger , the middle finger , and the

thumb now give the alignments of the x-, y-,

and z-axes, respectively.

• This is a so-called right-handed coordinate

system. z

x

y

Cartesian coordinate system

Experimental Physics - Mechanics - Vectors and Scalars 6

Vector components (2D)

y

x

yR

xR

sin

cos

RR

RR

y

x

x

y

yx

R

R

RRR

cos

sintan

22

– azimuthal angle

– polar angle

𝑅

Experimental Physics - Mechanics - Vectors and Scalars 7

Vector components (3D)

z

x

yR

xR

cos

sinsin

sincos

RR

RR

RR

z

y

x

R

R

R

R

RRRR

z

x

y

zyx

cos

cos

sintan

222

y

zR

𝑅

Experimental Physics - Mechanics - Vectors and Scalars 8

Unit vectors

z

y

x i

jk

𝑅

𝑅 = 𝑅𝑥𝑖 + 𝑅𝑦𝑗 + 𝑅𝑧𝑘

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Adding vectors by components

x 0 1 2 3 4 5 6 7 8 9

y

1

2

3

4

5

6

7

8

9

𝑅 = 𝐴 + 𝐵 = (𝐴𝑥 + 𝐵𝑥)𝑖 + (𝐴𝑦+𝐵𝑦)𝑗

𝐴 𝑅

𝐵

Experimental Physics - Mechanics - Vectors and Scalars 10

The scalar product

y

x

magnetic

field

𝐴

𝐵

𝐴 ∙ 𝐵 = 𝐴𝐵𝑐𝑜𝑠(𝜑)

𝑈 = −𝜇 ∙ 𝐵

𝐴 ∙ 𝐵 = 𝐴𝑥𝐵𝑥 + 𝐴𝑦𝐵𝑦 + 𝐴𝑧𝐵𝑧

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-3,0 -2,0 -1,0 0,0 1,0 2,0 3,0

-1,0

-0,5

0,0

0,5

1,0

cos

sin

Cos and Sin functions

Experimental Physics - Mechanics - Vectors and Scalars 12

The vector product

y

x

𝐴

𝐵

𝐶 = 𝐴 × 𝐵

𝐶 = 𝐴𝐵𝑠𝑖𝑛(𝜑)

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An example

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z y

x i

jk

jik

ikj

kji

ˆˆˆ

ˆˆˆ

ˆˆˆ

Some properties of vector product

0ˆˆ

0ˆˆ

0ˆˆ

kk

jj

ii

jki

ijk

kij

ˆˆˆ

ˆˆˆ

ˆˆˆ

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Some properties of vector product

Anticommutative:

y

x

Distributive over addition:

𝐴

𝐵

𝐴 × 𝐵 = −𝐵 × 𝐴

𝐴 × 𝐵 + 𝐶 = 𝐴 × 𝐵 + 𝐴 × 𝐶

𝐴 × 𝐵 = 𝐴𝑦𝐵𝑧 − 𝐴𝑧𝐵𝑦 𝑖 +

𝐴𝑧𝐵𝑥 − 𝐴𝑥𝐵𝑧 𝑗 +

𝐴𝑥𝐵𝑦 − 𝐴𝑦𝐵𝑥 𝑘

Experimental Physics - Mechanics - Vectors and Scalars 16

There are scalar and vector quantities.

Vectors can be added geometrically, but is more

straightforward in a component form.

The scalar components of a vector are its projections

to the axes of a Cartesian coordinate system.

Unit vectors are dimensionless, unit

vectors pointing along axes of a right-handed

coordinate system.

Two different types of vector products:

the scalar (dot) and vector (cross) products.

To remember!