Vectors

05/27/2023 1
Topic
Vectors
Presented By
Irfan Ali Bozdar
Lecturer
Department of Information Technology,

Shaheed Benazir Bhutto University, Shaheed
Benazirabad
05/27/2023 Irfan Ali Bozdar 2
Definition of a Vector
Vectors are quantities which have both magnitude (size) and direction.
They are used in navigation, engineering, science, economics, etc.
Geometrically, we can picture a vector as a directed line segment, whose
length is the magnitude of the vector and with an arrow indicating the
direction. The direction of the vector is from its tail to its head.

Scalar
A scalar or scalar quantity in physics is a physical quantity
that can be described by a single element of a number field
such as a real number, often accompanied by units of
measurement. A scalar is usually said to be a physical
quantity that only has magnitude and no other characteristics.
A scalar quantity is usually depicted by a number , numerical
value , or a magnitude , but no direction.
.

Scalar vs Vector

Vectors in Plane
A vector in the plane is represented
geometrically by a directed line segment whose
initial point is the origin and whose terminal
point is the point ( x1 , x2 ) as shown in given
Figure. This vector is represented by the same
ordered pair used to represent its terminal
point.
x  ( x1 , x2 )
x1 and x2
That is, The coordinates are called the components of the vector x. Two
vectors in the plane u  (u1 , u2 ) and v  (v1 , v2 ) are equal if and only if a u1  v1 and u 2  v 2

The term vector derives from the Latin word vectus,
meaning “to carry.” The idea is that if you were to carry
something from the origin to the point ( x1 , x2 ) the trip
could be represented by the directed line segment from
(0,0) to ( x1 , x2 ) .Vectors are represented by lowercase
letters set in boldface type (such as and u, v, w and x).

We denote vectors using bold face as in a or b.
Especially when writing by hand where one can not
easily write in bold face, then we will sometimes
 
denote vectors using arrows a or b , We denote the
magnitude of the vector a by ∥a∥. When we want to
refer to a number and stress that it is not a vector, we
can call the number a scalar. We will denote scalars
with italics, as in a or b

Applications
The common thread joining Linear Algebra with
many subfields of computer science is its vast power
as a tool for analyzing, understanding, and
manipulating information. The key for this is the use
of vectors as a means for representing, storing, and
manipulating data.

Documents: There are many possible ways to represent a written
document, one of them is to describe the document with a vector where
each entry corresponds to one unique word in the document’s language
(the English language, for instance, uses over 170-thousand words), and
the value stored at each entry is the number of times the word appears
in the document.

Images (and video): A single image (or a single frame from a movie)
consists of a set of pixels. Each pixel stores either brightness (for
grayscale images) or RGB colour data. We can store the data for each
pixel into consecutive entries in a vector. If we have an 32 × 32 grayscale
image such as the picture shown below, we will end up with a vector 1024
entries long containing the brightness values for each pixel. The
conversion works both ways: We can take any 1024 entry vector and
display its values as a 32 × 32 image. Thus, every possible 32 × 32
image is a point in a 1024-dimensional space.

Sound (music or speech): Sound data is made of a sequence of
values representing the amplitude (volume) of the sound over
time. These values can be stored sequentially into a vector. One
second of CD quality music would be represented by a vector
with 44, 100 values.

A great example would be how airplane pilots receive
instructions to land at airports. During a visual approach, the
Air Traffic Control instructs pilots to fly a particular
heading(direction) for a certain distance(magnitude). This is
exactly what a vector quantity is- something that has a
magnitude and direction. This is why the Air Traffic
Controllers might sometimes use the phrase " expect vectors
for the visual approach..." when the plane nears the airport.

Example: Vectors in the Plane
Use a directed line segment to represent each vector in the plane.
(a) u = (2, 3 ) and v = (-1, 2)

The first basic vector operation is vector addition. To add two vectors in the plane, add
their corresponding components. That is the sum of u and v is the vector.
u + v  (u1  u2 )  (v1  v2 )  (u1  v1 , u2  v2 )
Geometrically, the sum of two vectors in the plane is represented as the diagonal of
a parallelogram having u and v as its adjacent sides, as shown in above Figure. In
the next example on next slide, one of the vectors you will add is the vector (0, 0)
called the zero vector. The zero vector is denoted by 0.
Adding Two Vectors in the Plane
Find the sum of the following vectors.
(a ) u  (1, 4), v  (2,  2)  u  v = (1+ 2, 4 + (-2)) = (3, 2)
(b) u  (3,  2), v  (3, 2)  u  v = (3 + (-3),  2 + 2) = (0, 0)
(c) u  (2, 1), v  (0, 0)  u  v = (2 + 0, 1+ 0) = (2, 1)
Graphical representation of each sum

Operations with Vectors in the Plane
Provided with u = (3, 4) and v = (-2, 5) find each vector.
1 1 5
(a) v   2, 5    1, 
2 2  2
(b) u  v =  3, 4    2, 5    5,  1
1 1  5  13 
(c ) vu =  2, 5    3, 4    1,    3, 4    2, 
2 2  2  2 
Graphical representation of these vector operations.

Vector addition and scalar multiplication share many properties with matrix
addition and scalar multiplication. The ten properties listed in the next
theorem play a fundamental role in linear algebra. In fact, in the next section
you will see that it is precisely these ten properties that have been abstracted
from vectors in the plane to help define the general notion of a vector space.

William Rowan Hamilton (1805–1865) is considered to be
Ireland’s most famous mathematician. His work led to the
development of modern vector notation. We still use his i, j, and
k notation today.


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05/27/2023 1

Department of Information Technology,

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consists of a set of pixels. Each pixel stores either brightness (for

pixel into consecutive entries in a vector. If we have an 32 × 32 grayscale

display its values as a 32 × 32 image. Thus, every possible 32 × 32

image is a point in a 1024-dimensional space.

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(a) u = (2, 3 ) and v = (-1, 2)

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Graphical representation of each sum

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Graphical representation of these vector operations.

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