6.3 Vectors Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 1 Students will: Represent vectors as directed line segments. Write the

6.3 Vectors

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 1

Students will:Represent vectors as directed line

segments.Write the component form of

vectors.Perform basic vector operations

and represent vectors graphically.Write vectors as linear

combinations of unit vectors.Find the direction angles of

vectors.Use vectors to model and solve

real-life problems.

Do you remember?

• Which Trig Law should you use with each given information:(Law of Sines/Cosines?)

SAS ASA

SSS SSA

AAS



A ball flies through the air at a certain speed and in a particular direction. The speed and direction are both important quantities of the ball. The velocity is a vector quantity since it has both a magnitude and a direction.

Vectors are used to represent velocity, force, tension, and many other quantities.

A vector is a quantity with both a magnitude and a direction.


A quantity with magnitude and direction is represented by a directed line segment PQ.

Two vectors, u and v, are equal if the line segments representing them are parallel and have the same length or magnitude.And are in the same direction. u

v

P

QInitial point Terminal

point

The vector v = PQ is the set of all directed line

segments of length ||PQ|| which are parallel to PQ.

The Distance Formula:



Example 1 Equivalent Directed Line Segments

• Let u be the directed line segment from P=(0,0) to Q=(3,2)• Let v be the directed line segment from R=(1,2) to S=(4,4)

Show that u = v (Check magnitude and slope)


A vector with initial point (0, 0) is in standard position and is represented uniquely by its terminal point (v1, v2).

x

y

(v1, v2)

Components of v

P

Q

If point P is the origin and point Q = (5, 6), then v = 5, 6 .

The component form of vector v is written 1 2, .v v


If v is a vector with initial point P = (p1 , p2) and terminal point Q = (q1 , q2), then

x

y

1. The component form of v is

1 1 2 2 1 2= , , .PQ q p q p v v ��

v

2. The magnitude (or length) of v is

2 2 2 21 1 2 2 1 2= .q p q p v v v

P Q

(p1, p2)

(q1, q2)v

Try #1 page 417


Example 2a:Find the component form and magnitude of vector v that has initial point (2, 2) and terminal point (–1, 4).

The magnitude of v is

2 2= 3 2 9 4 13 3.61. v

x

y

-2 2

2 P = (2, 2)

Q = (–1, 4)

The component form of v is 3.61 units v

v =

v =

v

h

Example 2b Finding the Component Form of a Vector

Find the component form and magnitude of the vector v that has initial point (4,7) and terminal point (-1,5)

Try #11 page 417Initial pt. (-3,-5) Terminal pt. (5,1)

Scalar MultiplicationTwo basic vector operations are scalar multiplication and vector addition.

Scalar multiplication is the product of ku=k< u1,u2>

=< ku1, ku2>


Scalar multiplication is the product of a scalar, or real number, times a vector.

Example: Given vector = 4 ,, 2u find –2u.

4, 2 2 8, 42 u

x-4 4-8

y

4

-48, 4 –2u

1 2 1 2Let = , and , , and let be a scalar.u u v v ku v

1 2 1 2 = , = , k k u u ku kuu

The product of –2 and u gives a vector twice as long as and in the opposite direction of u.

4, 2

u

Vector AdditionTo add two vectors position them (without changing their length or direction) Position the vectors so that the initial point of one is located at the terminal point of the other.This technique is called the parallelogram law for vector additionThe vector u + v is often called the resultant of vector addition.


and = 4, 52 ,= 3, vuExample: Given vectorsfind u + v and u – v .

Example 3a Vector Addition and Subtraction

1 1 2 2= , u v u v u v 1 1 2 2= , u v u v u v


Try #19 page 418u = <4,2> , v = <7,1>

a. u+v

b. u-v

c. 2u-3v

d. v+4u


Unit VectorsA unit vector in the direction of v is a vector, u, that has a magnitude of 1 and the same direction as v.

1= unit vector = =

vu vv v

Divide v by its length to obtain a unit vector.

Example: Find a unit vector in the direction of v = –1, 1.

2 2

1, 1 =

1 1

vv

1= 1, 12

2 21 1= , = , 2 22 2

unit vector in the direction of v


Example #4 Finding a Unit Vector

Find a unit vector in the direction of v = <-2,5>

Verify that the result has a magnitude of 1

1= unit vector = =

vu vv v


Try #28 page 418 < 3,-4 >


Standard Unit VectorsThe unit vectors <1,0> and <0,1> are called standard unit

vectors and are denoted by

i = <1,0> and j = <0,1>

*NOTE: The letter i is written in bold to distinguish the difference it from imaginary i.

v = v v

v 1 ,0 v

1

1 2

,

,

2

0 1

The scalars and are the horizontal and vertical components of v

The linear combination of the vectors i and j is i + j

v 1 v 2v 1 v 2


Example #5 Writing a Linear Combination of Unit Vectors

Let u be the vector with initial point (2,5) and terminal point (-1,3). Write u as a linear combination of the standard vectors i and j.


Example #6 Vector OperationsLet u = -3i + 8j and v = 2i – j.

Find 2u – 3v


Direction AnglesIf u is a unit vector such that is the angle (measured counterclockwise) from the positive x-axis to u, the terminal point of u lies on the unit circle and

= , = cos , sin = cos + sin .x y u i j

is the direction angle of the vector u.

y

x

1

1

–1

– 1

(x, y)

y = sin

x = cos u

Continued.


Direction Angles continuedIf v = ai + bj is any vector that makes an angle with the positive x-axis, then it has the same direction as u and

cos= , sin v v

= + co s ns .i v i v j

The direction angle for v is determined by

ctan .

ossinsincos

ba

vv

Example: The direction angle of u = 3i +3j is3tan 1.3

ba

45


Example 7 Finding Direction Angles of Vectors

Find the direction angle of each vector

a.) 3i + 3j b.) 3i – 4j


Try #55 page 418

Homework

Day 1

p. 417-418 #1-3,5,11-12,15-16,19-20,28-29

Day 2

p. 418-420 #41-42,47-48,51-52,54-55,63,67, 69,73,75,81,82


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6.3 Vectors Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 1 Students will: Represent vectors as directed line segments. Write the