6.3 vectors in the plane day 1 2015 copyright © by houghton mifflin company, inc. all rights...

6.3Vectors in the Plane

Day 1 2015

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

A ball flies through the air at a certain speed and in a particular direction. The speed and direction are the velocity 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 with initial point P and terminal point Q.

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

u

v

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

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

P

Q

Vector Representation by Directed Line Segments

Let u be represented by the directed line segment from P = (0,0) to Q = (3,2), and let v be represented by the directed line segment from R = (1,2) to S = (4,4). Show that u = v.

1 2 3 4

4

3

2

1

P

QR

S

u

v

Using the distance formula, show that u and v have the same length.Show that their slopes are equal.

132414

130203

22

22

v

u

Slopes of u and v are both

3

2


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

For example, the scalar 3 times v results in the vector 3v, three times as long and in the same direction as v.

v

3v

v

The product of - and v gives a vector half as long

as and in the opposite direction to v. 2

1

2

1- v


Vector Addition

To add vectors u and v:

1. Place the initial point of u at the terminal point of v.

2. Draw the vector with the same initial point as v and the same terminal point as u.

uv

v + u

v u

vu


Vector Subtraction as adding the opposite.

To subtract vectors u and v: u vAdd the opposite of v to u: u +( v)

1. Place the initial point of v at the terminal point of u.

2. Draw the vector u v from the initial point of u to the terminal point of v.

vu

-v

u

u v

-v

u


Vector Subtraction

To subtract vectors u and v:

1. Place the initial point of v at the initial point of u.

2. Draw the vector u v from the terminal point of v to the terminal point of u.

vu

v

u

v

u

u v


A vector with initial point (0, 0) is in standard position and is represented uniquely by its terminal point (v1, v2). This is the component form of a vector v, written as .

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

1. The component form of v is

v = q1 p1, q2 p2

2. The magnitude (or length) of v is

||v|| =2

222

11 )()( pqpq

x

y(v1, v2)

x

y

P (p1, p2)Q (q1, q2)

1 2,v v


Remember that to write a vector in component form: v = q1 p1, q2 p2

Use terminal point – initial point.


Example:

Find the component form and magnitude of the vector v with initial point P = (3, 2) and terminal point Q = (1, 1).

The magnitude of v is

||v|| = = = 5.2 522 )3()4(

Component form: 4,3v

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

Let P = (4, -7) = (p1, p2) and Q = (-1, 5) = (q1, q2).

Then, the components of v = are given by 21,vv

Thus, v = 5, 12

The length of v is 1316912)5( 22 v


Example: If u = PQ, v = RS, and w = TU with P = (1, 2),

Q = (4, 3), R = (1, 1), S = (3, 2), T = (-1, -2), and U = (1, -1),

determine which of u, v, and w are equal. (solution follows.)

Calculate the component form for each vector:

u = 4 1, 3 2 = 3, 1

v = 3 1, 2 1 = 2, 1

w = 1 (-1), 1 (-2) = 2, 1

Therefore v = w but v = u and w = u./ /

Two vectors u = u1, u2 and v = v1, v2 are equal if and only if u1 = v1 and u2 = v2 .


Operations on Vectors in the Coordinate Plane

Let u = , v = , and let c be a scalar.

1. Scalar multiplication cu =

2. Addition u + v =

3. Subtraction u v =

1 1,x y2 2,x y

1 1,cx cy

1 2 1 2,x x y y

1 2 1 2,x x y y


x

y

u + v

x

y

Examples: Given vectors u = and v = Find -2u, u+v, and u-v Try these on your own.

-2u =

u + v = + = u v = =

(4, 2)u

(-8, -4) -2u

x

y

(6, 7)(2, 5)

(4, 2)v

u

(2, 5)

(4, 2)v

u

(2, -3)

u v

4,2 2,5

2 4,2 8, 4

4,2 2,5 6,7 4,2 2,5 2, 3

Vector Operations

Ex. Let v = and w = . Find the following vectors. a. 2v b. w – v

5,2 4,3

-2 2

6

10

8

4

2

-2-4

10,42 v

v

2v

1 2 3 4

4

3

2

1

5-1

w

5, 1w v

-v

w - v

v

v

1

v

vu = unit vector

Ex: Find a unit vector in the direction of v =

v

v

2,5

2 2 5 2

2,5

1

29 2,5

2

29,

5

29

A unit vector is a vector whose magnitude = 1. In many vector applications it is useful to find a unit vector that has the same direction as a given vector v. To do this, divide v by its length to obtain:


You try:

a) Find a unit vector in the direction of 3, 4

b) Find the magnitude of the unit vector you just found.

3 4,

5 5

1

Homework 6.3 Day 1

• Pg.417 1-35 odd


Vectors in the PlaneDay 2 2015

Digital Lesson


The unit vectors and are called the standard unit vectors. i represents one unit of horizontal movement and j represents one unit of vertical movement. Any vector can be represented by what is called a linear combination of unit vectors.

1,0i 0,1j

Example: Vector can be represented as a linear combination of unit vectors by rewriting it as v= 2i-6j.

2, 6v

Standard unit vectors

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 unitvectors of i and j.

-2 2 4

6

4

2

-2

-4

-6

-8

(2, -5)

u

(-1, 3)

Solution

3,8u

ji 83

-2 2

6

10

8

4

2

-2-4

Graphically,it looks like… -3i

8j

Writing a Linear Combination of Unit Vectors

Let u be the vector with initial point (1, -7) and terminal point(-1, 2).Write u as a linear combination of the standard unitvectors i and j.

Begin by writing the component form of the vector u.

2,9u 2 9u i j


Example: Find the vector v with the given magnitude and the same direction as u.

3, 4, 4v u 3 2 3 2,

2 2v


You try: Find the vector v with the given magnitude and the same direction as u.

10, 2 3v u i j 20 13 30 13

13 13v i j

Vector Operations

Let u = -3i + 8j and let v = 2i - j. Find 2u - 3v.

2u - 3v = 2(-3i + 8j) - 3(2i - j)

= -6i + 16j - 6i + 3j

= -12i + 19 j


x

y

Finding Direction AnglesThe direction angle of a vector v is the angle formed by the positive half of the x-axis and the ray along which v lies.

x

y

vθ v

θ

x

y

v

x

y

(x, y)

If v = 3, 4 , then tan = and = 53.13.

3

4

If v = x, y , then tan = . x

y

• Find the magnitude and direction angle of the vector:


3 3u i j 3 2, 45u

• Find the magnitude and direction angle of the vector:


3 4v i j 5, 306.87v

• Find the direction angle of the vector:


3 4v i j 5, 126.87v

• Find the direction angle of the vector:


3 4v i j 5, 233.13v


Writing Vectors using Direction Angles

If u is a unit vector such that is the angle (measured counter-clockwise) from the positive x-axis to u, the terminal point of u lies on the unit circle and you have:

, cos ,sin cos sinu x y i j

To indicate a length other than 1, multiply the unit vector by a magnitude.

The vector has a magnitude of 5 in the direction of 30 degrees.

5 cos30,sin 30v

cos ,sinv v

• Find the component form of v given its magnitude and the angle it makes with the positive x-axis:


3 and v is in the direction of 3i+4jv

3 cos53.13 ,sin 53.13v

Homework 6.3 Day 2

• Pg.418 37-61 odd


• Find the direction angle and write the component form using it:


4 and v is in the direction of -5i-2jv

4 cos 201.8 ,sin 201.8v

Vectors in the PlaneDay 3 2015

(use class notes handout)

HWQAdd vectors u and v.

u= 2i-j v = -i+j

Explain how you found the sum.

u+v = 1i+0j = I

Add two vectors by adding the horizontal components (x) , and the vertical (y) components.

The result is a new vector called the resultant of u and v.


• The sum of two or more vectors is called the resultant of the vectors.

• Find the resultant of u and v.

Resultant Vectors (sum of vectors)

3, 30uu

4, 60vv

resultant vector: r = 4.6,4.96

Applications


1.A ship leaving port sails for 75 miles in a bearing N 55 E. Find the magnitude of the vertical and horizontal components of this distance vector.

A ship leaving port sails for 75 miles at a bearing N 55 E. Find the magnitude of the vertical and horizontal components of this distance vector.

75cos35 61.44miles 75sin 35 43.02miles

Distance Vector: 75 cos35 ,sin 35

Applications



Find the component form of the vector that represents the velocity of a plane descending at a speed of 100 mph. @ an angle 30 degrees below horizontal.

1.Find the component form of the vector that represents the velocity of a plane descending at a speed of 100 mph. @ an angle of 30 degrees below horizontal.

100 cos 210 ,sin 210v

3 1100 ,

2 2v

50 3, 50v

2 2

50 3 50v 100

Applications



An airplane is traveling at a speed of 500 mph. with an air bearing of 330 degrees, at a fixed altitude, with negligible wind velocity. As the airplane reaches a certain point, it encounters a wind blowing with a velocity of 70 mph. in the direction Find :

a. The resultant speed. b. The direction of the airplane.

.

45 .N E

Resultant Speed

522.5v mph

Direction of the Plane

112.6

True Direction = 337.4

Applications



A piling for a high-rise building is pushed by two bulldozers at exactly the same time. One bulldozer exerts a force of 1550 pounds in a westerly direction. The other bulldozer pushes the piling with a force of 3050 pounds in a northerly direction. What is the magnitude of the resultant force upon the piling?

What is the direction of the resulting force upon the piling?

2 21550 3050 3421.25r

1 3050tan 116.94

1550

You Try



The Shanghai World Finance Center building in Shanghai, China, is 1508 feet tall. Suppose that a piling for building is pushed by two bulldozers at exactly the same time. One bulldozer exerts a force of 900 pounds in an easterly direction. The other bulldozer pushes the piling with a force of 2150 pounds in a northerly direction. What is the magnitude of the resultant force upon the piling?

What is the direction of the resulting force upon the piling?

2 2900 2150 2330.77r

1 2150tan 67.29

900

Homework 6.3 Day 3

• Pg.418 63-65 odd, 69-75odd ,81,82


6.3 vectors in the plane day 1 2015 copyright © by houghton mifflin company, inc. all rights...

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