experimental physics ep1 mechanics - vectors …...experimental physics - mechanics - vectors and...
TRANSCRIPT
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 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
Experimental Physics - Mechanics - Vectors and Scalars 5
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
Experimental Physics - Mechanics - Vectors and Scalars 9
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
Experimental Physics - Mechanics - Vectors and Scalars 11
-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)
Experimental Physics - Mechanics - Vectors and Scalars 14
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
Experimental Physics - Mechanics - Vectors and Scalars 15
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!