vectors in plane objectives of this section graph vectors find a position vector add and subtract...
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Vectors in Plane
Objectives of this Section
• Graph Vectors
• Find a Position Vector
• Add and Subtract Vectors
• Find a Scalar Product and Magnitude of a Vector
• Find a Unit Vector
• Find a Vector from its Direction and Magnitude
• Work With Objects in Static Equilibrium
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 of the vector.
P
Q
Initial Point
Terminal Point
Directed line segmentPQ
.point the topoint for the distance theis
segment line directed theof magnitude The
QP
PQ
. to from is ofdirection The QPPQ
If a vector v has the same magnitude and the same direction as the directed line segment PQ, then we write
PQv
The vector v whose magnitude is 0 is called the zero vector, 0.
v w if they have the same magnitude and direction.
Two vectors v and w are equal, written
vw
v w
v
wv w
Initial point of v
Terminal point of w
Vector Addition
Vector addition is commutative.
v w w v Vector addition is associative.
u v w u v w
v 0 0 v v
v v 0
Multiplying Vectors by Numbers
If is a scalar (a real number) and is a vector,
the is defined as
v
scalar product v1.
.
If > 0, the product is the vector
whose magnitude is times the magnitude
of and whose direction is the same as
v
v v
2.
.
If < 0, the product is the vector
whose magnitude is times the magnitude
of and whose direction is opposite that of
v
v v
3. . If = 0 or if , then v 0 v 0
Properties of Scalar Products
0 1 1v 0 v v v v
v v v v w v w
v v
Use the vectors illustrated below to graph each expression.
v
w
u
v w
v w
wv - and 2
v
2v
w
w
2v w
2v
w
If is a vector, we use the symbol to
represent the of
v v
magnitude v.
vv
vv
0vv
v
v
(d)
(c)
ifonly and if 0 b)(
0 (a)
thenscalar, a is if and vector a is If
A vector for which is called a
.
u u
unit vector
1
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. If v is a vector with initial point at the origin O and terminal point at P = (a, b), then
v i j a b
ai
bj
a
P = (a, b)
v = ai
+ bjb
The scalars a and b are called components of the vector v = ai + bj.
ectorposition v the toequal is
then, If .,point terminal
and origin, y thenecessarilnot ,,
point initialth vector wia is that Suppose
21222
111
v
v
v
PPyxP
yxP
v i j x x y y2 1 2 1
.4,3 and 1,2 if
vector theofector position v theFind
2121
PPPPv
v i j x x y y2 1 2 1
v i j 3 2 4 1( )
v i j 5 3
P1 2 1 ,
P2 3 4 ,
5 3,
O
v = 5i + 3j
Equality of Vectors
Two vectors v and w are equal if and only if their corresponding components are equal. That is,
If and +
then if and only if and
v i j w i j
v w
a b a b
a a b b1 1 2 2
1 2 1 2
,
.
Let and be two
vectors, and let be a scalar. Then,
v i j w i j a b a b1 1 2 2
v w i j
v w i j
v i j
v
a a b b
a a b b
a b
a b
1 2 1 2
1 2 1 2
1 1
12
12
If and , find
(a) (b)
v i j w i j
v w v w
3 2 4
(a) v w i j i j 3 2 4
3 4 2 1i j
i j3 (b) v w i j i j 3 2 4
3 4 2 1i j 7i j
If and , find
(a) 2 (b)
v i j w i j
v w v
3 2 4
3
(a) 2 3 2 3 2 3 4v w i j i j
6 4 12 3i j i j
6 7i j
(b) 2v i j 3 3 22 2
13
Unit Vector in Direction of v
For any nonzero vector v, the vector
uvv
is a unit vector that has the same direction as v.
Find a unit vector in the same direction as v = 3i - 5j.
v 3 52 2( ) 9 25 34
uvv
i j 3 534
334
534
i j
3 3434
5 3434
i j