Vectors Quantities that have magnitude but not direction are called scalars Ex Area volume temperature time etc Quantities such as force acceleration velocity or displacement that ID: 510219
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Slide1
12.2
VectorsSlide2
Quantities that have
magnitude but not direction are called scalars.Ex: Area, volume, temperature, time, etc.
Quantities such as force, acceleration, velocity or displacement that have direction as well as magnitude are represented by directed line segments, called vectors.
A
B
initial
point
terminalpoint
The
length
of the vector is calledthe magnitude and is denoted by
DefinitionsSlide3
A
vector is in standard position if the initial point is at the origin.
x
y
The
component form
of this vector is:
Vectors
are
equivalent
if they have the same length and direction.
then the
component form of
is:
If
are initial and terminal points of a vector,
P
Q
(
c,d
)
(
a,b
)
v
(a-c, b-d)
xSlide4
P
Q
(-3,4)
(-5,2)
The component form of
is:
v
(-2,-2)
The
magnitude is
ExampleSlide5
i
and
j
If
then
v
is a
zero vector
:
are called the
standard basis vectors
.
The
magnitude
of
is:
If
then
v
is a
unit vector
.Slide6
and
If
then
v
is a
zero vector
:
are called the
standard basis vectors
.
The
magnitude
of
is:
If
then
v
is a
unit vector
.
Vectors in Space Slide7
Vector sum:
Vector difference
Scalar Multiplication
:
Negative
(opposite):
Vector
v
is parallel to
u
if and only if
v =
k
u
for some
k.
Vector OperationsSlide8
v
v
u
u
u+v
u
+
v
is the
resultant vector
.
v
v
u
u
u-v
u
-
v
is the
resultant vector
.
Parallelogram LawSlide9
A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60
o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?
N
E
ApplicationSlide10
A Boeing 727 airplane, flying due
east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?
N
E
uSlide11
A Boeing 727 airplane, flying due east at 500mph in still air, encounters a
70-mph tail wind acting in the direction of 60o north of east
. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?
N
E
v
u
60
oSlide12
A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60
o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?
N
E
v
u
We need to find the magnitude and direction of the
resultant vector
u + v
.
u+vSlide13
N
E
v
u
The component forms of
u
and
v
are:
u+v
500
70
Therefore:
and:
The new ground speed of the airplane is about 538.4 mph, and its new direction is about 6.5
o
north of east.Slide14
i
j
v
is called a
linear combination
of i and
j
Any vectors can be written uniquely
in terms of standard basis vectors :
v
(measured counterclockwise) with the positive
x
-axis
then
v
can be written as
If
v
is
any nonzero vector that makes an angle
Linear CombinationSlide15
v
is called a
linear combination of
i, j and k
Standard basis vector notation
Linear Combination in SpaceSlide16
1) Find the unit vector in the direction of
v
2) Determine whether the points are collinear:
3) Show that the following points form the vertices of a parallelogram:
Examples