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What is a vector?
Physics/Physical Science
• A study of motion will involve the
introduction of a variety of quantities which
are used to describe the physical world.
• Examples of such quantities include
distance, displacement, speed, velocity,
acceleration, force, mass, momentum,
energy, work, power, etc. All these
quantities can by divided into two
categories - vectors and scalars.
Vectors vs. Scalars
• The difference between scalars and
vectors
• If a quantity has only a size, it is called a
scalar. Time and temperature are
examples of scalars. Mass, too, is an
example of a scalar.
• If a quantity has a size and a direction, it is
called a vector and can be symbolized, or
drawn, as an arrow. Velocity is an
example of a vector.
What is a Vector?
• A vector quantity is a quantity which is fully
described by both magnitude and
direction.
• The emphasis of this unit is to understand
some fundamentals about vectors and to
apply the fundamentals in order to
understand motion and forces which occur
in two dimensions.
• Examples of vector quantities which have
been previously discussed include
displacement, velocity, acceleration, and
force. Each of these quantities are unique
in that a full description of the quantity
demands that both a magnitude and a
direction are listed.
Do Not Write
• For example, suppose your teacher tells
you "A bag of gold is located outside the
classroom. To find it, displace yourself 20
meters." This statement may provide
yourself enough information to pique your
interest; yet, there is not enough
information included in the statement to
find the bag of gold.
Do Not Write
• The displacement required to find the bag
of gold has not been fully described. On
the other hand, suppose your teacher tells
you "A bag of gold is located outside the
classroom. To find it, displace yourself
from the center of the classroom door 20
meters in a direction 30 degrees to the
west of north."
Do Not Write
• This statement now provides a complete
description of the displacement vector - it
lists both magnitude (20 meters) and
direction (30 degrees to the west of north)
relative to a reference or starting position
(the center of the classroom door). Vector
quantities are not fully described unless
both magnitude and direction are listed.
Displacement
• To test your understanding of this
distinction, consider the motion depicted in
the diagram below. A physics teacher
walks 4 meters East, 2 meters South, 4
meters West, and finally 2 meters North.
• Even though the physics teacher has walked a total
distance of 12 meters, her displacement is 0 meters.
During the course of her motion, she has "covered 12
meters of ground" (distance = 12 m). Yet when she is
finished walking, she is not "out of place" - i.e., there is
no displacement for her motion (displacement = 0 m).
• Displacement, being a vector quantity,
must give attention to direction. The 4
meters east is canceled by the 4 meters
west; and the 2 meters south is canceled
by the 2 meters north. Vector quantities
such as displacement are direction aware.
• Vector quantities are often represented by
scaled vector diagrams. Vector diagrams
depict a vector by use of an arrow drawn
to scale in a specific direction. Vector
diagrams were introduced and used in
earlier units to depict the forces acting
upon an object.
• An example of a scaled vector diagram is
shown below. Observe that there are
several characteristics of this diagram
which make it an appropriately drawn
displacement vector diagram.
.
• Criteria of a vector diagram:
• a scale is clearly listed
• a vector arrow (with arrowhead) showing a specified
direction.
• the magnitude of 20 m and the direction is (30 degrees
West of North).
• The magnitude of a vector in a scaled
vector diagram is depicted by the length of
the arrow. The arrow is drawn a precise
length in accordance with a chosen scale.
• For example, the diagram below shows a
vector with a magnitude of 20 miles. Since
the scale used for constructing the
diagram is 1 cm = 5 miles, the vector
arrow is drawn with a length of 4 cm. That
is, 4 cm x (5 miles/1 cm) = 20 miles.
• In conclusion, vectors can be represented by
use of a scaled vector diagram. On such a
diagram, a vector arrow is drawn to represent
the vector. The arrow has an obvious tail and
arrowhead. The magnitude of a vector is
represented by the length of the arrow. A scale
is indicated (such as, 1 cm = 5 miles) and the
arrow is drawn the proper length according to
the chosen scale. The arrow points in the
precise direction
Practice Problems
What is the magnitude and
direction?
Vector Addition
• A variety of mathematical operations can
be performed with and upon vectors. One
such operation is the addition of vectors.
Two vectors can be added together to
determine the result (or resultant).
• Recall in our discussion of Newton's laws
of motion, that the net force experienced
by an object was determined by computing
the vector sum of all the individual forces
acting upon that object. That is the net
force was the result (or resultant) of
adding up all the force vectors.
• Observe
the following summations of two
:
force vectors
These rules for summing vectors were
applied to free-body diagrams in order to
determine the net force (i.e., the vector
sum of all the individual forces). Sample
applications are shown in the diagram
below.
• The task of summing vectors will be extended to
more complicated cases in which the vectors are
directed in directions other than purely vertical
and horizontal directions. For example, a vector
directed up and to the right will be added to a
vector directed up and to the left. The vector
sum will be determined for the more complicated
cases shown in the diagrams below.
Resultants
• The resultant is the vector sum of two or
more vectors. It is the result of adding two
or more vectors together. If displacement
vectors A, B, and C are added together,
the result will be vector R. As shown in the
diagram, vector R can be determined by
the use of an accurately drawn, scaled,
vector addition diagram.
• Any vector directed in two dimensions can
be thought of as having an influence in two
different directions. That is, it can be
thought of as having two parts. Each part
of a two-dimensional vector is known as a
component.
• If Fido's dog chain is stretched upward and
rightward and pulled tight by his master,
then the tension force in the chain has two
components - an upward component and
a rightward component.
• If the single chain were replaced by two
chains. with each chain having the
magnitude and direction of the
components, then Fido would not know
the difference.
What are the components of the
picture?
• The process of determining the magnitude
of a vector is known as vector resolution.
The two methods of vector resolution
which we will examine are:
• the parallelogram method
• the trigonometric method
• The two vectors at right angles to each
other, that add up to a given vector are
known as the components (Ax, & Ay).
• The diagram shows that the vector is first drawn to
scale in the indicated direction; a parallelogram is
sketched about the vector; the components are
labeled on the diagram; and the result of measuring
the length of the vector components and converting
to m/s using the scale. (NOTE: because different
computer monitors have different resolutions, the
actual length of the vector on your monitor may not
be 5 cm.)
• As the 60-Newton tension force acts
upward and rightward on Fido at an angle
of 40 degrees, the components of this
force can be determined using
trigonometric functions.
What is a projectile?
• A projectile is an object upon which the
only force acting is gravity.
• A projectile is any object which once
projected or dropped continues in motion
by its own inertia and is influenced only by
the downward force of gravity.
• Thus, the free-body diagram of a projectile
would show a single force acting
downwards and labeled force of gravity (or
simply Fgrav). Regardless of whether a
projectile is moving downwards, upwards,
upwards and rightwards, or downwards
and leftwards, the free-body diagram of
the projectile is still as depicted in the
diagram at the right
• Their belief is that forces cause motion;
and if there is an upward motion then
there must be an upward force. They
reason, "How in the world can an object be
moving upward if the only force acting
upon it is gravity?" Such students do not
believe in Newtonian physics
• A force is not required to keep an object in
motion. A force is only required to maintain
an acceleration. And in the case of a
projectile that is moving upward, there is a
downward force and a downward
acceleration. That is, the object is moving
upward and slowing down.
• Consider a cannonball shot horizontally
from a very high cliff at a high speed. And
suppose for a moment that the gravity
switch could be turned off such that the
cannonball would travel in the absence of
gravity? What would the motion of such a
cannonball be like? How could its motion
be described?
If the monkey lets go of the tree the moment that the
banana is fired, then where should she aim the banana
cannon? To ponder this question, first consider a scenario
in which there is no gravity acting on either the banana or
the monkey. What would be the path of the banana? Would
the banana hit the monkey?