Chapter 7: Vectors and the Geometry of Space Section 7.1 Vectors in the Plane Written by Dr. Julia Arnold Associate Professor of Mathematics Tidewater

Chapter 7: Vectors and the Geometry of Space

Section 7.1Vectors in the Plane

Written by Dr. Julia Arnold

Associate Professor of Mathematics

Tidewater Community College, Norfolk Campus, Norfolk, VA

With Assistance from a VCCS LearningWare Grant

In this first lesson on vectors, you will learn:

• Component Form of a Vector

• Vector Operations;

• Standard Unit Vectors;

• Applications of Vectors.

What is a vector?Many quantities in geometry and physics can be characterized by a single real number: area, volume, temperature, mass and time. These are defined as scalar quantities.

Quantities such as force, velocity, and acceleration involve both magnitude and direction and cannot be characterized by a single real number.To represent the above quantities we use a directed line segment.

What is a directed line segment?

First let us look at a directed line segment:

P

QThis line segment has a beginning, (the dot) and an ending (the arrow point).

We call the beginning point the “initial point” . Here we have called it P. The ending point (arrow point) is called the “terminal point” and here we have called it Q.

The vector is the directed line segment and is denoted by

PQIn some text books vectors will be denoted by bold type letters such as u, v, or w.

However, we will denote vectors the same way you will denote vectors by writing them with an arrow above the letter.

PQv

It doesn’t matter where a vector is positioned. All of the following vectors are considered equivalent.

because they are pointing in the same direction and the line segments have the same length.

How can we show that two vectors and are equivalent? u v

Suppose is the vector with initial point (0,0) and terminal point (6,4), and is the vector with initial point (1,2) and terminal point (7,6) .

vu

x

y

v

uSince a directed line segment is made up of its magnitude (or length) and its direction, we will need to show that both vectors have the same magnitude and are going in the same direction. Looks verify but are not proof.

How can we show that two vectors and are equivalent? u v

To show that the two vectors have the same direction we compute the slope of the lines.(0,0) and (6,4) Since they are also equal we (1,2) and (7,6) . Conclude the vectors are equal

v

2625216362617u 22

The symbol we use to denote the magnitude of a vector is what looks like double absolute value bars.

Thus represents the magnitude or length of the vector . v

2625216360406v 22

Both vectors have the same length, verified by using the distance formula.

32

64

1726

32

64

0604

uv

What is standard position for a vector in the plane?

Since all vectors of the same magnitude and direction are considered equal, we can position all vectors so that their initial point is at the origin of the Cartesian coordinate system. Thus the terminal point would represent the vector.

21 v,vv Would be the vector whose terminal point would be (v1,v2) and initial point (0,0)

The notation is referred to as the component form of v.

21 v,vv

v1 and v2 are called the components of v.

If the initial point and terminal point are both (0,0) then we call this the zero vector denoted as .0

Here is the formula for putting a vector in standard position:

If P(p1,p2) and Q(q1,q2) represent the initial point and terminal point respectively of a vector, then the component form of the vector PQ is given by:

22

21

222

211 vvpqpqPQ

212211 v,vpq,pqPQ

And the length is given by:

Special Vectors

If represents the vector v in standard position from P(0,0) to Q(v1,v2) and if the length of v,

21 v,vv

1v Then is called a unit vector.v

The length of a vector v may also be called the norm of v.

If then v is the zero vector . 0v 0

Vector Operations:

Now we need to define vector addition and scalar multiplication.

We will start with addition and look at the geometric interpretation.

x

y

v u

Move the first vector into standard position.

Vector Operations:



x

y

v u

Vector Operations:



x

y

v u

Vector Operations:



x

y

Move the second vector so that its initial point is at the terminal point of the first vector.

v

u

Vector Operations:



x

y


v

u

Vector Operations:



x

y


v

u

Vector Operations:



x

y


v

u

Vector Operations:



x

y


v

u

Vector Operations:



x

y


v

u

Vector Operations:



x

y


v

u

Vector Operations:



x

y


v

u

Vector Operations:


x

y

v

u

The result or the resultant vector is the one with initial point the origin and the terminal point at the endpoint of vector v.

2,1vu

1,112,21v

1,2u

v Is written in standard position.

See that the resultant vector can be found by adding the components of the vectors, u and v.

Vector Operations:


x

y

v

u

Notice, that if vector v is moved to standard position. The resultant vector becomes the diagonal of a parallelogram.

Vector Operations:


x

y

v

u

Notice, that if vector v is moved to standard position. The resultant vector becomes the diagonal of a parallelogram.

If and

2211 vu,vuvu

21 u,uu 21 v,vv

then the vector sum of u and v is

Next we look at a scalar multiple of a vector, 21 uk,kuuk

Example: Suppose we have a vector 3,2 that we double.

Geometrically, that would mean it would be twice as long, but the direction would stay the same. Thus only the length is affected.

If

then

ukuuk)uu(k

ukukkukuuk

uk,kuuk

22

21

22

21

2

22

221

222

21

21

If and

2211 vu,vuvu

21 u,uu 21 v,vv


If and k is a scalar then 21 uk,kuuk

Since -1 is a scalar, the negative of a vector is the same as multiplying by the scalar -1. So,

21 u,uu

Example: The negative of the vector would become3,2

3,2 Making the terminal point in the opposite direction of the original terminal point.

21 u,uu

x

y

3,2

3,2

If and

2211 vu,vuvu

21 u,uu 21 v,vv


21 uk,kuuk

The negative of is 21 v,vv 21 v,vv

Lastly, we examine the difference of two vectors:

2211 vu,vu)v(uvu

using the definition of the sum of two vectors and the negative of a vector.

If and k is a scalar then 21 u,uu

Geometrically, what is the difference? Let u and v be the vectors below.What is u – v?

x

y

3,3u

2,2v

vectors u and v are in standard position.

Now, create the vector -v

Geometrically, what is the difference? Let u and v be the vectors below.What is u – v?

x

y

3,3u

2,2v

vectors u and v are in standard position.

Now, create the vector -v

2,2v

Geometrically, what is the difference?

x

y

3,3u

2,2v

2,2v

1,5vu

Use the parallelogram principle to draw the sum of u - v

How do and relate to our parallelogram?

x

y

3,3u

2,2v

2,2v

1,5vu

5,1vu vu vu


x

y

3,3u

2,2v

2,2v

1,5vu

5,1vu vu vu


x

y

3,3u

2,2v

2,2v

1,5vu

5,1vu vu vu


x

y

3,3u

2,2v

2,2v

1,5vu

5,1vu vu vu


x

y

3,3u

2,2v

2,2v

1,5vu

5,1vu vu vu


x

y

3,3u

2,2v

2,2v

1,5vu

5,1vu vu vu

They are both diagonals of the parallelogram.

If and

2211 vu,vuvu

21 u,uu 21 v,vv


21 uk,kuuk The negative of is 21 v,vv 21 v,vv

2211 vu,vuvu

21 u,uu 21 v,vv If and

then the vector difference of u and v is

If and k is a scalar then 21 u,uu

Vector Properties of Operations

Let be vectors in the plane and let c, and d be scalars.wandv,u

The commutative property: uvvu

The associative property:

Additive Identity Property: uu00u

Additive Inverse Property: 0uu

Associative Property with scalars: )u(cdudc

Distributive Property: uducu)dc(

Distributive Property: vcuc)vu(c

wvuw)vu(

Also 0u0,u)u(1

ukuuk)uu(k

ukukkukuuk

uk,kuuk

22

21

22

21

2

22

221

222

21

21

If

then

The length of a scalar multiple of a vector is the length of the vector times the scalar as was shown earlier and here again.

Every non-zero vector can be made into a unit vector:0v,v

v

1

v

vu

Proof: First we will show that has length 1. u

1vv

1v

v

1u

Since is just a scalar multiple of ,

they are both going in the same direction. u v

The process of making a non-zero vector into a unit vector in the direction of is called the normalization of .v

v uv

Thus, to normalize the vector , multiply by the scalar . v vv

1

Example: Normalize the vector and show that the new vector has length 1.2,4

5220416242,4 22

Multiply 2,4 by 52

1

5

1,5

2

52

2,52

4

Now we will show that the normalized vector has length 1.

115

5

5

1

5

4

5

1,5

2

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

plane and are denoted by the symbols respectively.

jandi

1,0jand0,1i

Using this notation, we can write a vector in the plane in terms of the vectors

jandi as follows:

jviv1,0v0,1vv,vv 212121

jviv 21 is called a linear combination of . jandi

The scalars are called the horizontal and vertical components of 21 vandv

v respectively.

Writing a vector in terms of sin and cos .

Let be a unit vector in standard position that makes an angle with the x axis.

u

x

y

u

cos

sin

)sin,(cos

Thus

sin,cosu

Writing a vector in terms of sin and cos continued.

Let be a non-zero vector in standard position that makes an angle with the x axis.

v

Since we can make the vector v a unit vector by multiplying by the reciprocal of its length it follows that

jsinvicosvsin,cosvv

axisxthewith

makesvangletheiswherejsinicossin,cosv

v

Example: Suppose vector v has length 4 and makes a 30o angle with the positive x-axis. First we use the radian measure for

6

j2i32j2

14i

2

34

j6

sin4i6

cos46

sin,6

cos4v

Example 1:

Sample Problems

Find the component form of the vector v and sketch the vector in standard position with the initial point at the origin.

x

y

(3,1)

(-1,4)

Example 1:

Sample Problems

Find the component form of the vector v and sketch the vector in standard position with the initial point at the origin.

x

y

(3,1)

(-1,4)

3,414,31 <-4,3>

Example 2: Given the initial point <1,5> and terminal point <-3,6>, sketch the given directed line segment and write the vector in component form and finally sketch the vector in standard position.

Example 2: Given the initial point <1,5> and terminal point <-3,6>, sketch the given directed line segment and write the vector in component form and finally sketch the vector in standard position.

Solution: <-3-1,6-5>=<-4,1>

x

y

Example 3: Use the graph below to sketch v2u

x

y

uv

Example 3: Use the graph below to sketch v2u

x

y

uv

First double the length of v

Next move u

into standard position.

Now move v2

into standard position

Complete the parallelogram and draw the diagonal.

Example 4: Compute a2b3,b3,b2a,ba

For 5,4band,1,3a


For 5,4band,1,3a

613171817,18

17,182,615,121,3215,12a2b3

15,125,43b3

11,1110,81,35,421,3b2a

4,151,435,41,3ba

22

Solution:


For ji3band,j2ia


For ji3band,j2ia

255017ji7

ji7j4i2j3i9)j2i(2j3i9a2b3

j3i9)ji3(3b3

i5j2i6j2i)ji3(2j2ib2a

j3i4ji3j2iba

22

Solution:

Example 6: For each of the following vectors, a) find a unit vector in the same directionb) write the vector in polar coordinates i.e.

1.

2,5to1,2from

i4

j4i2

6,3

2.

3.

4.

sin,cosvv

Example 6: For each of the following vectors, a) find a unit vector in the same directionb) write the vector in polar coordinates

1.

10

10,

10

10310)b

10

10,

10

103

10

1,

10

31,3

10

1so,1019,1,312,25)a

2,5to1,2from

i4)b

ii44

14016)a

i4

j5

52i

5

552)b

j5

52i

5

5j

5

2i5

1j4i2

52

1so,5220164)a

j4i2

5

52,

5

553)b

5

52,

5

5

5

2,5

1

53

6,53

3so5345369)a

6,3

2.

3.

4.

sin,cosvv

Example 7: Suppose there are two forces acting on a skydiver: gravity at 150 lbs down and air resistance at 140 lbs up and 20 lbs to the right. What is the net force acting on the skydiver?

150

140

20

Example 7: Suppose there are two forces acting on a skydiver: gravity at 150 lbs down and air resistance at 140 lbs up and 20 lbs to the right. What is the net force acting on the skydiver?

150

140

20

The net force is the sum of the three forces acting on the skydiver.Gravity would be -150jAir Resistance would be 140jThe force to the right would be 20iThe sum would be which would be 10 pounds

down and 20 pounds to the right.

j10i20

Note: The 150 lbs represents the length of the vector. A unit vector pointing in the same direction is or -j. Thus in polar coordinates the vector would be 150(-j)=-150j

1,0

Example 8: Suppose two ropes are attached to a large crate. Suppose that rope A exerts a force of pounds on the crate and rope B exerts a force of . If the crate weighs 275 lbs., what is the net force acting on the crate? Based on your answer, which way will the crate move.

115,164177,177

A B

Example 8: Suppose two ropes are attached to a large crate. Suppose that rope A exerts a force of pounds on the crate and rope B exerts a force of . If the crate weighs 275 lbs., what is the net force acting on the crate? Based on your answer, which way will the crate move.

115,164177,177

A B

Solution: The weight of the crate combined with gravity creates a force of

-275j or<0,-275>.Adding the 3 vectors we get <13, 17 > 13 lbs right and 17 lbs up

Example 9: Find the horizontal and vertical components of the vector described.A jet airplane approaches a runway at an angle of 7.5o with the horizontal, traveling at a velocity of 160 mph.

Example 9: Find the horizontal and vertical components of the vector described.A jet airplane approaches a runway at an angle of 7.5o with the horizontal, traveling at a velocity of 160 mph.

7.5o

Solution: Remembering that speed is length of vector, we know that this vector is 160 miles in length. Using polar coordinates the vector is

88.20,63.1581305.,9914.160)5.7sin(,)5.7cos(160 oo

Example 10: A woman walks due west on the deck of a ship at 3 miles per hour. The ship is moving north at a speed of 22 miles per hour. Find the speed and direction of the woman relative to the surface of the water.N

E

S

W


E

S

WWoman3 mph

Ship 22mph

What angle is made by the woman relative to polar coordinates?

radians

What angle is made by the ship relative to polar coordinates?

2

radians

In unit vector terms this would be <-1,0>

In unit vector terms this would be <0,1>

Woman vector = 0,13 Ship vector= 1,022 Adding the two vectors22,322,00,31,0220,13


E

S

WWoman3 mph

Ship 22mph

Woman vector = 0,13 Ship vector= 1,022 Adding the two vectors22,322,00,31,0220,13

Notice this does not answer yet the question of speed and direction. Speed is vector magnitude and direction should be in degrees with a compass direction so how do we get that?

Example 10: A woman walks due west on the deck of a ship at 3 miles per hour. The ship is moving north at a speed of 22 miles per hour. Find the speed and direction of the woman relative to the surface of the water.

Woman3 mph

Ship 22mph

22,3Resultant Vector

Magnitude: mph2.22493484922,3

To find direction we need an angle:

o23.82...33333.7tan

...33333.73

22cossin

tan

1

When giving directions such as NW or SE you always begin with North or South and the angle is measured from either North or South. So is not the angle we would use to give the direction. We would use its complement which is 7.77o and say the woman is walking 22.2 mph in the direction N7.770W.

This angle

This would be our reference angle in Q3

Exercise 11. Given the vector with magnitude and direction following. Write the vector in component form.

Exercise 12. Given the vector in component form write the magnitude and direction of the vector with respect to N, NE, NW, S, SE, or SW direction.3i – 4j.

WN 030,2

Exercise 11. Given the vector with magnitude and direction following. Write the vector in component form.

Exercise 12. Given the vector in component form write the magnitude and direction of the vector with respect to N, NE, NW, S, SE, or SW direction.3i – 4j.

WN 030,2WN 030 Is in quadrant II with reference angle 60 degrees and from

the positive x axis 120 degrees, Thus the vector is

3,123

,21

2120sin,120cos2

Magnitude is: 525169

This vector is in Quadrant IV, (+,-) and

01 13.5334

tan

34

tan

Since 53.13 degrees would also be the reference angel between the vector and the positive x – axis, we would need to subtract from 90 degrees to find the angle between the vertical South and the vector for giving the direction of ESES 000 87.36)13.5390(

Your Homework for this section is in Blackboard under AssignmentsButton. Click on Assignment 7.1

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Chapter 7: Vectors and the Geometry of Space Section 7.1 Vectors in the Plane Written by Dr. Julia Arnold Associate Professor of Mathematics Tidewater