1. A scalar is a physical quantity that has only magnitude, such as mass, length, and time. A vector is a physical quantity that has both magnitude and direction, such as displacement, velocity, and force.
2. Vectors can be classified based on their orientation as parallel, anti-parallel, or perpendicular. They can also be added using geometric methods like the triangle law or analytical methods using components.
3. Multiplying a vector by a scalar multiplies the magnitude by the scalar value. Vector multiplication can produce a scalar using the dot product or a vector using the cross product. The dot product returns the magnitudes of the vectors and their cosine angle.
This presentation covers scalar quantity, vector quantity, addition of vectors & multiplication of vector. I hope this PPT will be helpful for Instructors as well as students.
This presentation covers scalar quantity, vector quantity, addition of vectors & multiplication of vector. I hope this PPT will be helpful for Instructors as well as students.
This upload is actually experimental, so sorry for the lost animations. This is my first post on SlideShare. Future presentations will take into account the loss of animation.
Also, I saw that the titles of all my slides got covered by something, so I'll never use this theme again. The titles of the slides are:
Slide 1: Vectors and Scalars
Slide 2: In this lecture, you will learn
Slide 3: What are vectors?
Slide 4: What are scalars?
Slide 5: A joke
Slide 6: A joke
Slide 7: What was that for?
Slide 8: What was that for?
Slide 9: Vectors
Slide 10: Geometric Representation
Slide 11: Vector Addition
Slide 12: Scalar Multiplication
Slide 13: The Zero Vector
Slide 14: The Negative of a Vector
Slide 15: Vector Subtraction
Slide 16: More Properties of Vector Algebra
Slide 17: Magnitude of a Vector
Slide 18: Vectors in a Coordinate System
Slide 19: Unit Vectors
Slide 20: Algebraic Representation of Vectors
Slide 21: Algebraic Addition of Vectors
Slide 22: Algebraic Multiplication of a Vector by a Scalar
Slide 23: Example 1
Slide 24: Example 2
Slide 25: A few words of caution
Slide 26: Problems
Motion of objects in physics are expressed by distance, displacement, speed, velocity, and acceleration which are associated with mathematical quantities called scalar and vector.
A vector has magnitude and direction. There is an algebra and geometry of vectors which makes addition, subtraction, and scaling well-defined.
The scalar or dot product of vectors measures the angle between them, in a way. It's useful to show if two vectors are perpendicular or parallel.
This upload is actually experimental, so sorry for the lost animations. This is my first post on SlideShare. Future presentations will take into account the loss of animation.
Also, I saw that the titles of all my slides got covered by something, so I'll never use this theme again. The titles of the slides are:
Slide 1: Vectors and Scalars
Slide 2: In this lecture, you will learn
Slide 3: What are vectors?
Slide 4: What are scalars?
Slide 5: A joke
Slide 6: A joke
Slide 7: What was that for?
Slide 8: What was that for?
Slide 9: Vectors
Slide 10: Geometric Representation
Slide 11: Vector Addition
Slide 12: Scalar Multiplication
Slide 13: The Zero Vector
Slide 14: The Negative of a Vector
Slide 15: Vector Subtraction
Slide 16: More Properties of Vector Algebra
Slide 17: Magnitude of a Vector
Slide 18: Vectors in a Coordinate System
Slide 19: Unit Vectors
Slide 20: Algebraic Representation of Vectors
Slide 21: Algebraic Addition of Vectors
Slide 22: Algebraic Multiplication of a Vector by a Scalar
Slide 23: Example 1
Slide 24: Example 2
Slide 25: A few words of caution
Slide 26: Problems
Motion of objects in physics are expressed by distance, displacement, speed, velocity, and acceleration which are associated with mathematical quantities called scalar and vector.
A vector has magnitude and direction. There is an algebra and geometry of vectors which makes addition, subtraction, and scaling well-defined.
The scalar or dot product of vectors measures the angle between them, in a way. It's useful to show if two vectors are perpendicular or parallel.
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Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much of that energy is released.
Cell utilize energy in the form of ATP.
The first step of respiration is called glycolysis. In a series of steps, glycolysis breaks glucose into two smaller molecules - a chemical called pyruvate. A small amount of ATP is formed during this process.
Most healthy cells continue the breakdown in a second process, called the Kreb's cycle. The Kreb's cycle allows cells to “burn” the pyruvates made in glycolysis to get more ATP.
The last step in the breakdown of glucose is called oxidative phosphorylation (Ox-Phos).
It takes place in specialized cell structures called mitochondria. This process produces a large amount of ATP. Importantly, cells need oxygen to complete oxidative phosphorylation.
If a cell completes only glycolysis, only 2 molecules of ATP are made per glucose. However, if the cell completes the entire respiration process (glycolysis - Kreb's - oxidative phosphorylation), about 36 molecules of ATP are created, giving it much more energy to use.
IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 – 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
WARNBURG EFFECT : cancer cells under aerobic (well-oxygenated) conditions to metabolize glucose to lactate (aerobic glycolysis) is known as the Warburg effect. Warburg made the observation that tumor slices consume glucose and secrete lactate at a higher rate than normal tissues.
10. Displacement from USA to China is 11600 km
in east
physical quantity magnitude
direction
USA China
11. A vector is a physical quantity that has
both a magnitude and a direction.
Examples:
• Position
• Displacement
• Velocity
• Acceleration
• Momentum
• Force
36. Magnitude of Resultant
P
O
B
A
C
Q R
M
θ
θ cos θ =
AM
AC
AM = AC cos θ⇒
In ∆CAM,
OC2 = OM2 + CM2
OC2
= (OA + AM)2
+ CM2
OC2 = OA2 + 2OA × AM + AM2
+ CM2
OC2 = OA2 + 2OA × AM + AC2
R2 = P2 + 2P × Q cos θ + Q2
In ∆OCM,
R = P2 + 2PQ cos θ + Q2
OC2
= OA2
+ 2OA × AC cos θ
+ AC2
37. Direction of Resultant
P
O A
C
Q R
M
α θ
In ∆CAM,
cos θ =
AM
AC
AM = AC cos θ⇒
sin θ =
CM
AC
CM = AC sin θ⇒
In ∆OCM,
tan α =
CM
OM
tan α =
CM
OA+AM
tan α =
AC sin θ
OA+AC cos θ
tan α =
Q sin θ
P + Q cos θ
B
38. Case I – Vectors are parallel ( 𝛉 = 𝟎°)
P Q R
R = P2 + 2PQ cos 0° + Q2
R = P2 + 2PQ + Q2
R = (P + Q)2
+ =
Magnitude: Direction:
R = P + Q
tan α =
Q sin 0°
P + Q cos 0°
tan α =
0
P + Q
= 0
α = 0°
39. Case II – Vectors are perpendicular (𝛉 = 𝟗𝟎°)
P Q
R
R = P2 + 2PQ cos 90° + Q2
R = P2 + 0 + Q2
+ =
Magnitude: Direction:
tan α =
Q sin 90°
P + Q cos 90°
=
Q
P + 0
α = tan−1
Q
P
P
Q
R = P2 + Q2
α
40. Case III – Vectors are anti-parallel (𝛉 = 𝟏𝟖𝟎°)
P Q R
R = P2 + 2PQ cos 180° + Q2
R = P2 − 2PQ + Q2
R = (P − Q)2
− =
Magnitude: Direction:
R = P − Q
tan α =
Q sin 180°
P + Q cos 180°
= 0
α = 0°If P > Q:
If P < Q: α = 180°
41. Unit vectors
A unit vector is a vector that has a magnitude of exactly
1 and drawn in the direction of given vector.
• It lacks both dimension and unit.
• Its only purpose is to specify a direction in space.
A
𝐴
42. • A given vector can be expressed as a product of
its magnitude and a unit vector.
• For example A may be represented as,
A = A 𝐴
Unit vectors
A = magnitude of A
𝐴 = unit vector along A
44. Resolution of a Vector
It is the process of splitting a vector into two or more vectors
in such a way that their combined effect is same as that of
the given vector.
𝑡
𝑛
A 𝑡
A 𝑛
A
52. Magnitude & direction from components
A = A 𝑥 𝑖 + A 𝑦 𝑗 + A 𝑧 𝑘
A = A 𝑥
2
+ A 𝑦
2
+ A 𝑧
2
Magnitude:
Direction:
𝛼 = cos−1
A 𝑥
A
𝛽 = cos−1
A 𝑦
A
𝛾 = cos−1
A 𝑧
A
A
A 𝑧
𝛾
𝛽
𝛼
A 𝑥
A 𝑦
53. Adding vectors by components
A = A 𝑥 𝑖 + A 𝑦 𝑗 + A 𝑧 𝑘
B = B 𝑥 𝑖 + B 𝑦 𝑗 + B 𝑧 𝑘
Let us have
then
R = A + B
R = A 𝑥 𝑖 + A 𝑦 𝑗 + A 𝑧 𝑘
+ B 𝑥 𝑖 + B 𝑦 𝑗 + B 𝑧 𝑘
R = (A 𝑥 + B 𝑥) 𝑖 + (A 𝑦 + B 𝑦) 𝑗
+ (A 𝑧 + B 𝑧) 𝑘
R 𝑥 𝑖 + R 𝑦 𝑗 + R 𝑧 𝑘 = (A 𝑥+B 𝑥) 𝑖
+ (A 𝑦+B 𝑦) 𝑗 + (A 𝑧+B 𝑧) 𝑘
R 𝑥 = (A 𝑥 + B 𝑥)
R 𝑦 = (A 𝑦 + B 𝑦)
R 𝑧 = (A 𝑧 + B 𝑧)
55. Multiplying a vector by a scalar
• If we multiply a vector A by a scalar s, we get a
new vector.
• Its magnitude is the product of the magnitude
of A and the absolute value of s.
• Its direction is the direction of A if s is positive
but the opposite direction if s is negative.
57. Multiplying a vector by a vector
• There are two ways to multiply a vector by a
vector:
• The first way produces a scalar quantity and
called as scalar product (dot product).
• The second way produces a vector quantity and
called as vector product (cross product).
59. Examples of scalar product
W = F ∙ s
W = Fs cos θ
W = work done
F = force
s = displacement
P = F ∙ v
P = Fv cos θ
P = power
F = force
v = velocity
60. Geometrical meaning of Scalar dot product
A dot product can be regarded as the product
of two quantities:
1. The magnitude of one of the vectors
2.The scalar component of the second vector
along the direction of the first vector
61. Geometrical meaning of Scalar product
θ
A ∙ B = A(B cos θ) A ∙ B = (A cos θ)B
A
θ
A
A cos θ
B
B
62. Properties of Scalar product
1
The scalar product is commutative.
A ∙ B = AB cos θ
B ∙ A = BA cos θ
A ∙ B = B ∙ A
63. Properties of Scalar product
2
The scalar product is distributive over
addition.
A ∙ B + C = A ∙ B + A ∙ C
64. Properties of Scalar product
3
The scalar product of two perpendicular
vectors is zero.
A ∙ B = AB cos 90 °
A ∙ B = 0
65. Properties of Scalar product
4
The scalar product of two parallel vectors
is maximum positive.
A ∙ B = AB cos 0 °
A ∙ B = AB
66. Properties of Scalar product
5
The scalar product of two anti-parallel
vectors is maximum negative.
A ∙ B = AB cos 180 °
A ∙ B = −AB
67. Properties of Scalar product
6
The scalar product of a vector with itself
is equal to the square of its magnitude.
A ∙ A = AA cos 0 °
A ∙ A = A2
68. Properties of Scalar product
7
The scalar product of two same unit vectors is
one and two different unit vectors is zero.
𝑖 ∙ 𝑖 = 𝑗 ∙ 𝑗 = 𝑘 ∙ 𝑘 = (1)(1) cos 0 ° = 1
𝑖 ∙ 𝑗 = 𝑗 ∙ 𝑘 = 𝑘 ∙ 𝑖 = (1)(1) cos 90 ° = 0
69. Calculating scalar product using components
A = A 𝑥 𝑖 + A 𝑦 𝑗 + A 𝑧 𝑘
B = B 𝑥 𝑖 + B 𝑦 𝑗 + B 𝑧 𝑘
Let us have
then
A ∙ B = A 𝑥 𝑖 + A 𝑦 𝑗 + A 𝑧 𝑘
∙ (B 𝑥 𝑖 + B 𝑦 𝑗 + B 𝑧 𝑘)
A ∙ B = A 𝑥 𝑖 ∙ B 𝑥 𝑖 + B 𝑦 𝑗 + B 𝑧 𝑘
+ A 𝑦 𝑗 ∙ B 𝑥 𝑖 + B 𝑦 𝑗 + B 𝑧 𝑘
+ A 𝑧 𝑘 ∙ B 𝑥 𝑖 + B 𝑦 𝑗 + B 𝑧 𝑘
= A 𝑥B 𝑥 𝑖 ∙ 𝑖 + A 𝑥B 𝑦 𝑖 ∙ 𝑗 + A 𝑥B 𝑧 𝑖 ∙ 𝑘
+ A 𝑦B 𝑥 𝑗 ∙ 𝑖 + A 𝑦B 𝑦 𝑗 ∙ 𝑗 + A 𝑦B 𝑧 𝑗 ∙ 𝑘
+ A 𝑧B 𝑥 𝑘 ∙ 𝑖 + A 𝑧B 𝑦 𝑘 ∙ 𝑗 + A 𝑧B 𝑧 𝑘 ∙ 𝑘
= A 𝑥B 𝑥(1) + A 𝑥B 𝑦(0) + A 𝑥B 𝑧(0)
+ A 𝑦B 𝑥(0) + A 𝑦B 𝑦(1) + A 𝑦B 𝑧(0)
+ A 𝑧B 𝑥(0) + A 𝑧B 𝑦(0) + A 𝑧B 𝑧(1)
A ∙ B = A 𝑥B 𝑥 + A 𝑦B 𝑦 + A 𝑧B 𝑧
72. Examples of vector product
τ = r × F
τ = rF sin θ 𝑛
τ = torque
r = position
F = force
L = r × p
L = rp sin θ 𝑛
L = angular momentum
r = position
p = linear momentum
73. Geometrical meaning of Vector product
θ
A × B = A(B sin θ)
A
B
A × B = A sin θ B
θ
A
B
Asinθ
A × B = Area of parallelogram made by two vectors
74. Properties of Vector product
1
The vector product is anti-commutative.
A × B = AB sin θ 𝑛
B × A = BA sin θ (− 𝑛) = −AB sin θ 𝑛
A × B ≠ B × A
75. Properties of Vector product
2
The vector product is distributive over
addition.
A × B + C = A × B + A × C
76. Properties of Vector product
3
The magnitude of the vector product of two
perpendicular vectors is maximum.
A × B = AB sin 90 °
A × B = AB
77. Properties of Vector product
4
The vector product of two parallel vectors
is a null vector.
A × B = AB sin 0 ° 𝑛
A × B = 0
78. Properties of Vector product
5
The vector product of two anti-parallel
vectors is a null vector.
A × B = AB sin 180 ° 𝑛
A × B = 0
79. Properties of Vector product
6
The vector product of a vector with itself
is a null vector.
A × A = AA sin 0 ° 𝑛
A × A = 0
80. Properties of Vector product
7
The vector product of two same unit
vectors is a null vector.
𝑖 × 𝑖 = 𝑗 × 𝑗 = 𝑘 × 𝑘
= (1)(1) sin 0 ° 𝑛 = 0
81. Properties of Vector product
8
The vector product of two different unit
vectors is a third unit vector.
𝑖 × 𝑗 = 𝑘
𝑗 × 𝑘 = 𝑖
𝑘 × 𝑖 = 𝑗
𝑗 × 𝑖 = − 𝑘
𝑘 × 𝑗 = − 𝑖
𝑖 × 𝑘 = − 𝑗
83. Calculating vector product using components
A = A 𝑥 𝑖 + A 𝑦 𝑗 + A 𝑧 𝑘
B = B 𝑥 𝑖 + B 𝑦 𝑗 + B 𝑧 𝑘
Let us have
then
A × B = A 𝑥 𝑖 + A 𝑦 𝑗 + A 𝑧 𝑘
× B 𝑥 𝑖 + B 𝑦 𝑗 + B 𝑧 𝑘
A × B = A 𝑥 𝑖 × B 𝑥 𝑖 + B 𝑦 𝑗 + B 𝑧 𝑘
+ A 𝑦 𝑗 × B 𝑥 𝑖 + B 𝑦 𝑗 + B 𝑧 𝑘
+ A 𝑧 𝑘 × B 𝑥 𝑖 + B 𝑦 𝑗 + B 𝑧 𝑘
= A 𝑥B 𝑥 𝑖 × 𝑖 + A 𝑥B 𝑦 𝑖 × 𝑗 + A 𝑥B 𝑧 𝑖 × 𝑘
+ A 𝑦B 𝑥 𝑗 × 𝑖 + A 𝑦B 𝑦 𝑗 × 𝑗 + A 𝑦B 𝑧 𝑗 × 𝑘
+ A 𝑧B 𝑥 𝑘 × 𝑖 + A 𝑧B 𝑦 𝑘 × 𝑗 + A 𝑧B 𝑧 𝑘 × 𝑘
= A 𝑥B 𝑥(0) + A 𝑥B 𝑦( 𝑘) + A 𝑥B 𝑧(− 𝑗)
+ A 𝑦B 𝑥(− 𝑘) + A 𝑦B 𝑦(0) + A 𝑦B 𝑧( 𝑖)
+ A 𝑧B 𝑥( 𝑗) + A 𝑧B 𝑦(− 𝑖) + A 𝑧B 𝑧(0)
= A 𝑦B 𝑧 𝑖 − A 𝑧B 𝑦 𝑖 + A 𝑧B 𝑥 𝑗
− A 𝑥B 𝑧( 𝑗) + A 𝑥B 𝑦( 𝑘) − A 𝑦B 𝑥( 𝑘)
84. Calculating vector product using components
A × B = 𝑖 A 𝑦B 𝑧 − A 𝑧B 𝑦 − 𝑗 A 𝑥B 𝑧 − A 𝑧B 𝑥
+ 𝑘 (A 𝑥B 𝑦 − A 𝑦B 𝑥)
A × B =
𝑖 𝑗 𝑘
A 𝑥 A 𝑦 A 𝑧
B 𝑥 B 𝑦 B 𝑧