vectors in plane objectives of this section graph vectors find a position vector add and subtract...
TRANSCRIPT
![Page 1: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/1.jpg)
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
![Page 2: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/2.jpg)
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.
![Page 3: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/3.jpg)
P
Q
Initial Point
Terminal Point
Directed line segmentPQ
![Page 4: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/4.jpg)
.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
![Page 5: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/5.jpg)
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
![Page 6: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/6.jpg)
v
wv w
Initial point of v
Terminal point of w
Vector Addition
![Page 7: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/7.jpg)
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
![Page 8: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/8.jpg)
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
![Page 9: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/9.jpg)
Properties of Scalar Products
0 1 1v 0 v v v v
v v v v w v w
v v
![Page 10: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/10.jpg)
Use the vectors illustrated below to graph each expression.
v
w
u
![Page 11: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/11.jpg)
v w
v w
![Page 12: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/12.jpg)
wv - and 2
v
2v
w
w
![Page 13: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/13.jpg)
2v w
2v
w
![Page 14: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/14.jpg)
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
![Page 15: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/15.jpg)
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
![Page 16: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/16.jpg)
ai
bj
a
P = (a, b)
v = ai
+ bjb
![Page 17: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/17.jpg)
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
![Page 18: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/18.jpg)
.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
![Page 19: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/19.jpg)
P1 2 1 ,
P2 3 4 ,
5 3,
O
v = 5i + 3j
![Page 20: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/20.jpg)
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
,
.
![Page 21: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/21.jpg)
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
![Page 22: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/22.jpg)
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
![Page 23: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/23.jpg)
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
![Page 24: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/24.jpg)
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.
![Page 25: 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](https://reader036.vdocuments.net/reader036/viewer/2022062321/56649ef05503460f94c008c5/html5/thumbnails/25.jpg)
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