10.2 day 1: Vectors in the Plane

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003

Mesa Verde National Park, Colorado

Warning:

Only some of this is review.

Quantities that we measure that have magnitude but not direction are called scalars.

Quantities such as force, displacement or velocity that have direction as well as magnitude are represented by directed line segments.

A

B

initialpoint

terminalpoint

AB��������������

The length is AB��������������

A

B

initialpoint

terminalpoint

AB��������������

A vector is represented by a directed line segment.

Vectors are equal if they have the same length and direction (same slope).

A vector is in standard position if the initial point is at the origin.

x

y

1 2,v v

The component form of this vector is: 1 2,v vv

A vector is in standard position if the initial point is at the origin.

x

y

1 2,v v

The component form of this vector is: 1 2,v vv

The magnitude (length) of 1 2,v vv is:2 2

1 2v v v

P

Q

(-3,4)

(-5,2)

The component form of

PQ��������������

is: 2, 2 v

v(-2,-2) 2 2

2 2 v

8

2 2

If 1v Then v is a unit vector.

0,0 is the zero vector and has no direction.

Vector Operations:

1 2 1 2Let , , , , a scalar (real number).u u v v k u v

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

(Add the components.)

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

(Subtract the components.)

Vector Operations:

Scalar Multiplication:1 2,k ku kuu

Negative (opposite): 1 21 ,u u u u

v

vu

u

u+vu + v is the resultant vector.

(Parallelogram law of addition)

Any vector can be written as a linear

combination of two standard unit vectors.

,a bv

1,0i 0,1j

,a bv

,0 0,a b

1,0 0,1a b

a b i j

The vector v is a linear combination

of the vectors i and j.

The scalar a is the horizontal

component of v and the scalar b is

the vertical component of v.

The angle between two vectors is given by:

1 1 1 2 2cosu v u v

u v

This comes from the law of cosines.See page 524 for the proof if you are interested.

The dot product (also called inner product) is defined as:

1 1 2 2cos u v u v u v u v

Read “u dot v”

Example:

3,4 5,2

3 5 4 2 23

The dot product (also called inner product) is defined as:

1 1 2 2cos u v u v u v u v

This could be substituted in the formula for the angle between vectors (or solved for theta) to give:

1cos

u v

u v

Find the angle between vectors u and v:

2,3 , 2,5 u v

1cos

u v

u v

Example:

1 2,3 2,5cos

2,3 2,5

1 11cos

13 29

55.5

HORIZONTAL AND VERTICAL COMPONENTS

The horizontal and vertical components, of a vector u having magnitude u and direction angle theta are given by:

That is,

cos sina u b u

, cos , sinu a b u u

Application: Example 7

A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?

N

E

Application: Example 7

A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?

N

Eu

Application: Example 7

A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?

N

E

v

u

60o

Application: Example 7

A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?

N

E

v

u

We need to find the magnitude and direction of the resultant vector u + v.

u+v

N

E

v

u

The component forms of u and v are:

u+v

500,0u

70cos60 ,70sin 60v

500

70

35,35 3v

Therefore: 535,35 3 u v

538.4 22535 35 3 u v

and: 1 35 3tan

535 6.5

N

E

The new ground speed of the airplane is about 538.4 mph, and its new direction is about 6.5o north of east.

538.4

6.5o

We can describe the position of a moving particle by a vector, r(t).

tr

If we separate r(t) into horizontal and vertical components,

we can express r(t) as a linear combination of standard unit vectors i and j.

t f t g t r i j f t i

g t j

Most of the rules for the calculus of vectors are the same as we have used, except:

Speed v t

velocity vectorDirection

speed

t

tv

v

“Absolute value” means “distance from the origin” so we must use the Pythagorean theorem.