6.36.3 vectors in the plane. quick review quick review solutions
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Quick Review SolutionsTRANSCRIPT
6.36.3Vectors in the PlaneVectors in the Plane
Quick Review
-1
1. Find the values of and .
32. Solve for in degrees. sin11
3. A naval ship leaves Port Northfolk and averages 43 knots (nauticalmph) traveling for 3 hr on a bearing of 35 and then 4 h
x y
r on a courseof 120 . What is the boat's bearing and distance from Port Norfolkafter 7 hr.
Quick Review Solutions
-1
1. Find the values of and .
32. Solv
7.5, 7.5 3
64.8e for in degrees. sin 11
3. A naval ship leaves Port Northfolk and averages 43
x y
x y
knots (nauticalmph) traveling for 3 hr on a bearing of 35 and then 4 hr on a courseof 120 . What is the boat's bearing and distance from Port N
distance=orfolk
a 224.2; fter 7 bearin ghr. =84.9
What you’ll learn about• How to represent vectors as directed line segments• How to perform basic Vector Operations• How to write vectors as linear combinations of Unit
Vectors• How to find the Direction Angles of vectors• How to use vectors to model and solve real-life
problems… and whyThese topics are important in many real-world
applications, such as calculating the effect of the wind on an airplane’s path.
Directed Line Segment
Two-Dimensional Vector
A is an ordered pair of real numbers, denoted in as , . The numbers and are
the of the vector . The of the vector ,
a b a b
a b
two - dimensional vectorcomponent form
components standard representation
v
vis the arrow from the origin to the point ( , ).
The of is the length of the arrow and the of is the direction in which the arrow is pointing. The vector
= 0,0 , called the
a b
magnitude direction
0 ze
vv
, has zero length and no direction.ro vector
Initial Point(R), Terminal Point(S), Equivalent(P)
Magnitude
1 1 2 2
2 2
2 1 2 1
2 2
If is represented by the arrow from , to , , then
.
If , , then .
x y x y
v x x y y
a b a b
v
v v
Example Finding Magnitude of a Vector
Find the magnitude of represented by , where (3, 4) and
(5, 2).PQ P
Q
v
Example Finding Magnitude of a Vector
Find the magnitude of represented by , where (3, 4) and
(5, 2).PQ P
Q
v
2 2
2 1 2 1
2 2
5 3 2 ( 4)
2 10
x x y y
v
Vector Addition and Scalar Multiplication
1 2 1 2
1 1 2 2
Let , and , be vectors and let be a real number
(scalar). The (or ) is the vector, .
The and the vector is
u u v v k
u v u v
k k u
sum resultant of the vectors and
product of the scalar
u v
u vu v
k uu
1 2 1 2, , .u ku ku
Example Performing Vector Operations
Let 2, 1 and 5,3 . Find 3 . u v u v
Example Performing Vector Operations
Let 2, 1 and 5,3 . Find 3 . u v u v
3 3 2 , 3 1 = 6, 3
3 = 6, 3 5,3 6 5, 3 3 11,0
u
u v
Unit Vectors
A vector with || || 1 is a . If is not the zero vector10,0 , then the vector is a
|| || || ||.
unit vector
unit vector in the direction
of
u u vvu vv v
v
Example Finding a Unit Vector
Find a unit vector in the direction of 2, 3 . v
Example Finding a Unit Vector
Find a unit vector in the direction of 2, 3 . v
222, 3 2 3 13, so
1 2 32, 3 , 13 13 13
| |
| |
v
v
v
Standard Unit Vectors
The two vectors 1,0 and 0,1 are the standard
unit vectors. Any vector can be written as an expressionin terms of the standard unit vector:
,
,0 0,
1,0 0,1
a b
a b
a b
a b
i j
v
v
i j
Resolving the Vector If has direction angle , the components of can be computed
using the formula = | | cos , | | sin .
From the formula above, it follows that the unit vector in the
direction of is cos ,sin .| |
v v
v v v
vv uv
Example Finding the Components of a
Vector
Find the components of the vector with direction angle 120 andmagnitude 8.
v
Example Finding the Components of a
Vector
Find the components of the vector with direction angle 120 andmagnitude 8.
v
, 8cos120 ,8sin120
1 3 8 ,82 2
4,4 3
So 4 and 4 3.
a b
a b
v
Example Finding the Direction Angle of a Vector
Find the magnitude and direction angle of 2,3 .u
Example Finding the Direction Angle of a Vector
Find the magnitude and direction angle of 2,3 .u
2 2|| || 2 3 13Let be the direction angle of , then
2,3 13 cos , 13sin
2 13 cos56.3
uu
u
Velocity and SpeedThe velocity of a moving object is a
vector because velocity has both magnitude
and direction. The magnitude of velocity is
speed.