Vectors
GEOMETRIC AND ALGEBRAIC FORMS
Geometric Form
y A vector is a quantity that has both magnitude (length) and direction. A
vector is represented geometrically by a directed line segment. It has an
initial point and a terminal point.
y The length of the line segment represents the magnitude of the vector
and the arrowhead indicates the direction of the vector.
y If a vector has an initial point at the origin, it is in standard position.
The direction of the vector is the directed angle between the positive x
axis and the vector.
y
x
Equal vectors
y We denote vectors with a ray above the letter. Two vectors are equal if
and only if they have the same direction and the same magnitude.
a
b
d
c
a and c are equal since they have the same direction and the same
magnitude. b and d have different directions so they are not equal.
Addition of vectors
y The sum of two or more vectors is called the resultant. To show the
addition of two vectors, use the tip to tail method. Put the initial point
of the second vector on the tail of the first vector. Then draw the
resultant from the initial point of the first vector to the terminal point
of the second vector. (notice same size and direction)
p
q
q
p
p+q
Subtraction of vectors
y Two vectors are opposite if they have the same magnitude and opposite
directions.
p – q = p + -q
p
q
If
p
-q
p-q
then
Scalar Quantity
y A quantity with only magnitude is called a scalar quantity. Examples of
scalars include mass, length, time and temperature. The numbers used
to measure scalar quantities are called scalars.
y If
a
then 3a
Algebraic Form
y Vectors can be represented algebraically using ordered pairs of real
numbers. A vector does not need to be in standard position to be
expressed algebraically. It can be in 2 dimensional space with (x, y) or
in three dimensional space with (x, y, z). The same rules apply.
Representing a vector as an ordered pair
y The ordered pair is the difference in the x coordinates of the initial and
terminal points and the difference in the y coordinates of the initial and
terminal points.
y Example:
y Vector with initial point at (1,1) and terminal point at (4,3)
y
The ordered pair that represents this vector is
(4-1, 3-1) = (3, 2)
(x2-x1)²+(y2-y1)²
The magnitude =
So the magnitude of this vector is
13
In three dimensional space
y The ordered pair would by
(x -x , y -y , z -z ).
y The magnitude would by
(x -x )² +( y -y )² +( z -z )²
2
2
1
1
2
1
2
2
1
1
2
1
Operations of vectors
y When vectors are represented by ordered pairs, they
can be easily added, subtracted, or multiplied by a
scalar.
UNIT VECTOR
y A unit vector has a magnitude of one unit. A unit
vector in the direction of the positive x axis is
represented by i and a unit vector in the direction of
the positive y axis is represented by j . Any vector
w= (w1, w2) can be represented as w1i+w2j. In three
dimensional, the unit vector on the x-y-z axes are i,
j, k where i=(1,0,0) j=(0,1,0) and k=(0,0,1). So any
vector w in three dimensional space w = (w1,w2,w3)
can be represented as w= w1i + w2j +w3k
HW # 22
ySection 8-2-Pp. 497-499 #15-31 odds
ySection 8.3: Pg 503 #7-25 odd