Transcript 11.4 Vectors
8.4
Vectors
A
vector
is a quantity that has both magnitude and direction. Vectors in the plane can be represented by arrows.
The length of the arrow represents the
magnitude
of the vector.
The arrowhead indicates the
direction
the vector.
of
Initial Point
P
Directed line segment
Q
Terminal Point
The magnitude of the directed line segment
PQ
is the distance from point to the point
Q.
P
The direction of
PQ
is from
P
to
Q
. If a vector the same direction as the directed line segment
v
PQ
has the same magnitude and , then we write
v =
PQ
The vector
v
whose magnitude is 0 is called the
zero vector, 0.
Two vectors
v
and
w v
are
equal,
written
w
if they have the same magnitude and direction.
v w v
=
w
v
+
w w
Terminal point of
w
Initial point of
v
v
Vector addition is
commutative.
v + w = w + v
Vector addition is
associative.
v + (u + w) = (v + u) + w v + 0 = 0 + v =v v + (-v) = 0
Multiplying Vectors by Numbers
v 2v -v
Properties of Scalar Products
Use the vectors illustrated below to graph each expression.
w v u
v w
v + w
2v
v
w
2v and -w
-w
2v-w
2v -w
An algebraic vector
v
is represented as
v
= <
a, b
> where
a
and
b
are real numbers (scalars) called the
components
of the vector
v
.
If
v
= <
a, b
> is an algebraic vector with initial point at the origin
O
and terminal point
P
= (
a
,
b
), then called a
position vector
.
v
is
y O P =
(
a, b
)
x
The scalars
a
components
a, b
>.
and
b
are called of the vector
v
= <
Theorem Suppose that
v
is a vector with initial point
P
1 =(
x
1
, y
1 ), not necessarily the origin, and terminal point
P
2 =(
x
2
, y
2 ). If
v
=
P
1
P
2 , then
v
is equal to the position vector
Find the position vector of the vector
v
=
P
1
P
2 if
P
1 =(-2, 1) and
P
2 =(3,4).
P
1 =(-2, 1)
O
P
2 =(3,4).
Theorem Equality of Vectors Two vectors
v
and
w
are equal if and only if their corresponding components are equal. That is,
Let
i
denote a unit vector whose direction is along the positive
x
-axis; let
j
denote a unit vector whose direction is along the positive
y
-axis. Any vector
v
= <
a
,
b
> can be written using the unit vectors
i
and
j
as follows:
Theorem Unit Vector in Direction of
v
For any nonzero vector
v
, the vector
is a unit vector that has the same direction as
v
.
Find a unit vector in the same direction as
v
= 3
i
-
5
j