This document discusses vectors and their application to motion in two and three dimensions. It introduces vectors as representations of displacement that have both magnitude and direction. Displacement, velocity and acceleration can all be described as vectors, allowing their addition and subtraction using graphical vector addition. Vector components in Cartesian coordinates are also discussed. The document uses examples of particle motion to illustrate average and instantaneous velocity and acceleration in two dimensions and the concept of relative velocity.
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
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
To investigate the relation between the ratio of (i) output and input voltage and (ii)
number of turns in the secondary coil and primary coil of a self-designed transformer.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
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This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
1. Motion in Two and Three Dimensions:
Vectors
Physics 1425 Lecture 4
Michael Fowler, UVa.
2. Today’s Topics
• In the previous lecture, we analyzed the
motion of a particle moving vertically under
gravity.
• In this lecture and the next, we’ll generalize to
the case of a particle moving in two or three
dimensions under gravity, like a projectile.
• First we must generalize displacement,
velocity and acceleration to two and three
dimensions: these generalizations are vectors.
3. Displacement
• We’ll work usually in two dimensions—the three dimensional
description is very similar.
• Suppose we move a ball
from point A to point B on a
tabletop. This displacement
can be fully described by
giving a distance and a direction.
• Both can be represented by an arrow, the length some
agreed scale: arrow length 10 cm representing 1 m
displacement, say.
• This is a vector, written with an arrow : it has magnitude,
meaning its length, written , and direction.
r
| |
r
4. Displacement as a Vector
• Now move the ball a
second time. It is
evident that the total
displacement , the sum
of the two, called the
resultant, is given by
adding the two vectors
tip to tail as shown:
• Adding displacement
vectors (and notation!):
1
r
2
r
1 2
r r
+
5. Adding Vectors
• You can see that
• The vector represents
a displacement, like
saying walk 3 meters in a
north-east direction: it
works from any starting
point.
• Adding vectors :
1
r
2
r
1 2 2 1.
r r r r
+ = +
1
r
2
r
1
r
6. Subtracting Vectors
• It’s pretty easy: just ask,
what vector has to be
added to to get ?
• The answer must be
• To construct it, put the
tails of , together,
and draw the vector
from the head of to
the head of .
Finding the difference:
a
b
b a
−
a
b
a
b
a
b
b a
−
7. Multiplying Vectors by Numbers
• Only the length changes: the direction stays
the same.
• Multiplying and adding or subtracting:
a
2a
a
−
a
b
2 3
a b
−
8. Vector Components
• Vectors can be related to
the more familiar Cartesian
coordinates (x, y) of a point
P in a plane: suppose P is
reached from the origin by a
displacement
• Then can be written as
the sum of successive
displacements in the x- and
y-directions:
• These are called the
components of
• Define to be
vectors of unit length
parallel to the x, y axes
respectively. The
components are
.
r
r
ˆ ˆ
,
i j
ˆ
xi
ˆ
yj
ˆ ˆ
r xi yj
= +
.
r
ˆ ˆ
, .
xi yj
9. How Relates to (x, y)
• The length (magnitude)
of is
The angle between the vector
and the x-axis is given by:
• a
r
ˆ
xi
ˆ
yj
ˆ ˆ
r xi yj
= +
θ
r
2 2
r x y
= +
tan .
y
x
θ =
10. Average Velocity in Two Dimensions
average velocity = displacement/time
• A In moving from point to ,
the average velocity is in the
direction :
2 1
2 1
r r
v
t t
−
=
−
2
r
1
r
1
r
2 1
r r
−
2
r
2 1
r r
−
11. Instantaneous Velocity in Two Dimensions
• Note: is small, but
that doesn’t mean has
to be small— is small
too!
Defined as the average
velocity over a
vanishingly small time
interval : points in
direction of motion at
that instant:
0
lim
t
r dr
v
t dt
∆ →
∆
=
∆
1
r
r
∆
2
r
r
∆
t
∆
v
12. Average Acceleration in Two Dimensions
• Car moving along curving road:
Note that the velocity vectors tails must be
together to find the difference between them.
2 1
2 1
v v
a
t t
−
=
−
1
r
2
r
1
v
2
v
1
v
2
v
2 1
v v
−
14. Acceleration in Vector Components
Writing and matching:
as you would expect from the one-dimensional case.
2
2
dv d dr d r
a
dt dt dt dt
= = =
( ) ( )
, , ,
x y
a a a r x y
= =
2 2
2 2
,
x y
d x d y
a a
dt dt
= =
15. Clicker Question
A car is moving around a circular track
at a constant speed. What can you say
about its acceleration?
A. It’s along the track
B. It’s outwards, away from the center of the
circle
C. It’s inwards
D. There is no acceleration
16. Relative Velocity
Running Across a Ship
• A cruise ship is going north at 4 m/s through
still water.
• You jog at 3 m/s directly across
the ship from one side to the other.
• What is your velocity relative to the water?
17. Relative Velocities Just Add…
• If the ship’s velocity relative to the water is
• And your velocity relative to the ship is
• Then your velocity relative to the water is
• Hint: think how far you are displaced in one second!
1
v
2
v
1 2
v v
+