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Vectors in the PlaneObjectives:•Define a vector•What are the basic terms associated with vectors?
DefinitionA vector is a quantity that has magnitude (size) and direction. It is used to measure quantities, such as force, velocity and acceleration.
A vector is a displacement
A vector is a translation
Traveling northeast at a speed of 60 mph.
60 mph NE
Example
ExampleTraveling west at a speed of 30 mph.
30 mphwes
t
A vector is represented using a directed line segment.
v
DefinitionVectors that point in the opposite direction and have the same length are opposite vectors.
u vu v
2 2a b
DefinitionThe magnitude of a vector is represented by its length. If , then the magnitude is :
,,
a bu a b T
2 2 u a b
ab
DefinitionVectors may be multiplied by a scaler. A scaler is a real number. It has magnitude (size) only.u v
2 w v
12
u vw
Given rv
1r
is a scalarmultiple
ofv
Describe the effect on the vector when:
.
0 1r
0r
“stretches” the vector “shrinks” the vector “reflects” the vector through a point
DefinitionVectors are parallel if and only if they are scalar Multiples of each other.
Given Points A(3,2) B(6,4)C(4, -2) and D(2,-1)
v AB
and w CD
Express each as a translation1)v 2)w
3)2v 4) 3w5)v w 6)v w
Do Now:
Every vector consists of at least 2 component vectors:1 in the x direction and 1 in the y direction This vector is in standard position.Component Vectors
Aim: How do we represent vectors algebraically ?
Important IdeaA vector may be represented by subtraction of the ordered pairs.
u(2,3)(4,5)
(2,2)u2,2
ExampleFind the component form of the vector v and sketch the vector with its initial point at the origin.
(2,3)
(5,5)
(3,2)
3,2
DefinitionVector Addition:
,u v a c b d
,u a b
,v c dIf and then
u v is also
calledor resultant vector.
the vector sum
u
vu v
Try This2,8u
5, 3v If and ,
find u v
7,5
DefinitionVector Subtraction: If
,u a b and , then: ,v c d
,u v a c b d
Vector Subtraction:,u v a c b d
DefinitionScaler Multiplication:
,u a bIf and r is a real
,ru ra rb
number, then:
Try This2, 3u
If and r=2, find ru
4, 6
and sketch its graph in standard position.
Definition1,0i and 0,1j
are unit vectors. Any vector can be written as a scaler times i plus a scaler times j.
Example2,3 2,0 0,3
2 1,0 3 0,1
2 3i j
Try ThisWrite using unit vectors.
5, 3
5i-3j
10cos130 ,10sin130
6.43 , 7.66
A Force F of 10 Newtons acts at an angle of with the positive x-axis. Express F as a vector in component form.
o130
Given A(0,4) and B(6,1). Find the coordinates of point P that is 2/3 of the way from A to B.
NEXT DO NOW
Try This
Solution:OP OA AP
20,43
AB
20,4 6, 33
0,4 4, 2
4,2
Given A(8,4) and B(-1,7). Find the coordinates of point P that is 2/3 of the way from A to B.
Another One
O28,4328,4 9,33
8,4 6,2
2,6
P
OA AP
AB
Describing Position:, 0,3 2, 1 x y t
The position of an object at time t is given as,x y and 0,3 are standard
position vectorsWhere is the object at t=0? t=1 t=2? t = 3? t = 4?
t R(t)01234
, 0,3 2, 1 x y t
0,32,24,16,08, 1
, 0,3 2, 1 x y t
What is the speed of the Object?
What direction is the Object moving?
0 0 1 2, , ,x y x y t v v
Vector Equation of a lineGiven standard position vectors
and The equation of a line is:
,x y0 0,x y
0 0,x y
1 2,v vis the Initial Position is the Velocity or Direction
Vector 1 2,v v is the Speed
, 4,9 1, 2 x y tFlyWhat is the speed of the Fly?
What direction is the Fly movingHow are these Flies different?
, 4,9 2, 4 x y t, 4,9 1,2 x y t
Fly 2Fly 3
Can this spider catch any of the Flies? Explain.
, 1,1 1,1 x y t
An object moves with constant velocity. At t = 0 it is at A(5,3) and at t = 3 it is at B(−4,15). a) find its velocity b) find its speed c) Write a vector equation describing the objects movement.
Example