1 Kidoguchi, Kenneth

§8.4 Vectors

Objectives

xi

yj

zk

0rrt < 0

t > 0

1. Graph Vectors

2. Find a Position Vector

3. Add & Subtract Vectors

Algebraically

4. Find a Scalar Multiple

and the Magnitude of a

Vector

5. Find a Unit Vector

6. Find a Vector from Its

Direction and

Magnitude

7. Model with Vectors

§8.4 Vectors

Vectors and Vector Notations

The term vector is used by scientists

to indicate a quantity (such as

displacement or velocity or force) that

has both magnitude and direction.

A vector is often represented by an

arrow. The length of the arrow

represents the magnitude of the vector

and the arrow points in the direction of

the vector.

A vector must be denoted by printing

a letter in boldface (v) or by putting an

arrow above the letter റ𝑣 . A symbol

that is not boldface or has no arrow

above it is a scalar and NOT a vector.

A scalar is NEVER equal to a vector!

A

B

𝐯 = 𝐴𝐵

21 November 2017 2 Kidoguchi, Kenneth

𝐯 = v = റ𝑣 ≠ 𝑣

v is the vector that is

directed from A to B

§8.4 Vectors

Vector Geometry

If the vector 𝐮 = 𝐶𝐷 has the same

length (aka magnitude) and the

same direction as 𝐯 even though it

is in a different position we say that

𝐮 and 𝐯 are equivalent (or equal)

and we write 𝐮 = 𝐯.

The vector −𝐯 = 𝐵𝐴 has the same

length as 𝐯 but points in the

opposite direction.

The zero vector, denoted by 𝟎, has

length 0. It is the only vector with

no specific direction.

C

D

𝐮 = 𝐶𝐷

A

B−𝐯 = 𝐵𝐴

A

B

𝐯 = 𝐴𝐵

21 November 2017 3 Kidoguchi, Kenneth

§8.4 Vectors

Vector Addition

A

B

𝐯 = 𝐴𝐵Definition of Vector Addition:

If 𝐯 and 𝐰 are vectors positioned

so the initial point (tail) of 𝐰 is at

the terminal point (arrow head)

of 𝐯, then the sum 𝐯 + 𝐰 is the

vector from the initial point (tail)

of 𝐯 to the terminal point (head)

of 𝐰.

C

𝐰 = 𝐵𝐶

𝐯 +𝐰 = 𝐴𝐶

21 November 2017 4 Kidoguchi, Kenneth

§8.4 Vectors

Basis Vectors and Vector Notation

A

B

𝐯 = 𝐴𝐵

C

𝐰 = 𝐵𝐶

i

j

Basis vectors are unit vectors

(i.e. vectors of unit magnitude)

and specify direction. We will

use Ƹ𝑖 (pronounced i cap) a unit

vector along the positive x-axis

and Ƹ𝑗 a unit vector along the

positive y-axis .

ˆ ˆ2

2, 1

2, 1

2

1

2,1

i j

v

ˆ ˆ2

1, 2

1, 2

1

2

1, 2

i j

w

21 November 2017 5 Kidoguchi, Kenneth

Verboten !!!

§8.4 Vectors

Basis Vectors and Vector Notation

A

B

𝐯 = 𝐴𝐵

C

𝐰 = 𝐵𝐶

𝐯 +𝐰 = AC

i

j

The vector sum of 𝐯 and 𝐰 is the

sum of their components.

2 1

1 2

3

1

v w

21 November 2017 6 Kidoguchi, Kenneth

§8.4 Vectors

Vector Magnitude and Trigonometric Form

A

B

𝐯 = 𝐴𝐵

i

j

q

cos

sin

cos

sin

v

v

v

q

q

q

q

v

2 2

For

tan

a

b

v

a b

b

a

q

v

v v

2 2

2

2

2 2

2tan

2 4

v

q q

v

v v

2 2 cos / 4

2 2 sin / 4

cos / 42 2

sin / 4

v

21 November 2017 7 Kidoguchi, Kenneth

§8.4 Vectors

Graphing Vectors Geometrically

21 November, 2017 8 Kidoguchi, Kenneth

i

j

w

c) 2wh

b) vw

Use the vectors illustrated to

graph the following vectors.

a) vwv

h

m112_vectorAddition.mw.

21 November, 2017 9 Kidoguchi, Kenneth

§8.4 Vectors

Graphing Vectors Geometrically

vw

vw

wv

i

j

h

Use the vectors illustrated to

graph the following vectors.

vw

2wh

2w

v

c) 2wh

b) vw

a) vw

m112_vectorAddition.mw.

§8.4 Vectors

Position Vectors

10 Kidoguchi, Kenneth

Suppose that 𝐯 is a vector with initial point P1 = (x1, y1), not necessarily the

origin, and terminal point P2 = (x2, y2). If 𝐯 = 𝑃1𝑃2 , then 𝐯 is equal to the

position vector:

2 1 2 1

2 1 2 1

2 1

2 1

,

ˆˆ

x x y y

i x x k y y

x x

y y

v

§8.4 Vectors

Resolving a Vector into Rectangular Components

21 November, 2017 11 Kidoguchi, Kenneth

x

y

i

j

1 2 3-3 -2 -1 0

1

2

3

-1

-2

-3

= (-2, 3)

= (3, -1)

(x2, y2)

(x1, y1)

v

Alternate Notation:

2

2

1

1

xv

y

x

y

2 1pv p

2p

1p

21 November, 2017 12 Kidoguchi, Kenneth

§8.4 Vectors

Resolving a Vector into Rectangular Components

x

y

i

j

1 2 3-3 -2 -1 0

1

2

3

-1

-2

-3

= (-2, 3)

= (3, -1)

,

5

2 3 1

4

3

,

(x2, y2)

(x1, y1)

v

Alternate Notation:

2

2

1

1

xv

y

x

y

2 3

3 1

5

4

2 1pv p

1 12 2,xx y y

2p

1p

§8.4 Vectors

Vector Arithmetic

13 Kidoguchi, Kenneth

Let 𝐯 = 𝑎1 , 𝑏1 and 𝐰 = 𝑎2 , 𝑏2 be two vectors and let a be a

scalar. Then:

1 2 1 2

1 2 1 2

1 2 1 2

1 2 1 2

1 1

1 1

2 2

1 1

a a a a

b b b b

a a a a

b b b b

a a

b b

v a b

v w

v w

v

v v

§8.4 Vectors

A Unit Vector in the Direction of 𝐯

For any nonzero vector 𝐯, the vector

is a unit vector of unit magnitude and has the same direction as

vector 𝐯. Hence:

v

ˆ

v

v

v

u

v

v

v

𝐯

ෝ𝐮

ˆ ˆ ˆ v u u v uv

21 November, 2017 15 Kidoguchi, Kenneth

§8.4 Vectors

Resolving a Vector into Rectangular Components

vw

vw 2ˆ ˆ2 2,1

1i j

)a v

) 2d wh

wv

i

j )e vw

h)c h

Sketch the given vectors and

express them in component

form.

)b w1

ˆ ˆ 1, 11

i j

0ˆ2 0,2

2j

0 1 22

2 1 0

1

2

vw

)f vw

2wh

2w

w

v

m112_vectorAddition.mw.

21 November, 2017 16 Kidoguchi, Kenneth

When the river is still, a boat

crosses it with velocity vector

റ𝑣𝑠ℎ𝑖𝑝 = 4,0 km/hr. Today, the

river flows with a velocity

റ𝑣𝑟𝑖𝑣𝑒𝑟 = 0,−3 km/hr. If the

river is 20 km wide, present the

analysis to find the:

§8.4 Vectors

Example Vector Application

a) resultant velocity vector of the boat.

b) resultant speed of the boat.

c) time it will take the boat to get from one river bank to the other if the

river is 20 km wide.

d) total distance travelled by the boat.

20 kmRiv

er B

ank

1

Riv

er B

ank

2

vrivervship

ij

17 Kidoguchi, Kenneth

§8.4 Vectors

Example Vector Application: An Object in Static Equilibrium

18 Kidoguchi, Kenneth

A box of supplies that weighs

1000 pounds is suspended by

two cables attached to the

ceiling as shown. Present the

analysis to find the tensions in

the two cables.

§8.4 Vectors

Example Vector Application: True Aircraft Speed & Direction

19 Kidoguchi, Kenneth

A Boeing 737 aircraft maintains a constant airspeed

of 600 miles per hour headed due south. The jet

stream is 60 miles per hour in the northeasterly

direction.

a) Express 𝑣𝑎 the velocity 737 the relative to 𝑣𝑤the velocity of the jet stream in terms if Ƹ𝑖 and Ƹ𝑗.

b) Find റ𝑣 the velocity of the 737 relative to the

ground.

c) Find the actual speed and direction of the 737

relative to the ground.

§8.4 Vectors

Example Vector Application: Finding the Weight of a Piano

20 Kidoguchi, Kenneth

Two movers require a magnitude of

force of 350 pounds to push a piano

up a ramp inclined at an angle of 25°

from the horizontal. How much does

the piano weigh?