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Vectors and Scalars

A vector  has magnitude as

well as direction.

Some vector quantities:

displacement, velocity, force,

momentumA scalar  has only a magnitude.

Some scalar quantities: mass,

time, temperature

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Distance: A Scalar Quantity

 A scalar quantity:

Contains magnitudeonly and consists of anumber and a unit.

(20 m, 40 mi/h, 10 gal)

 A

B

Distance is the length of the actual pathtaken by an object.

s = 20 m

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Displacement— A Vector Quantity

 A vector quantity:

Contains magnitude

 AND direction, anumber, unit & angle.

(12 m, 300; 8 km/h, N)

 A

BD = 12 m, 20o 

• Displacement is the straight-lineseparation of two points in a specifieddirection.

q

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More about Vectors

•  A vector is represented on paper by an

arrow

1. the length represents magnitude

2. the arrow faces the direction of

motion

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A vector is a quantity that has both magnitude and

direction. It is represented by an arrow. The length of

the vector represents the magnitude and the arrow

indicates the direction of the vector.

Two vectors are equal if they have the same direction andmagnitude (length).

Blue and orange 

vectors have

same magnitudebut different

direction.

Blue and green 

vectors have

same directionbut different

magnitude.

Blue and purple

vectors have

same magnitudeand direction so

they are equal.

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6

Magnitude of a Vector

• The magnitude of a vector is a positive number (with units!)

that describes its size.

• Example: magnitude of a displacement vector is its length.

• The magnitude of a velocity vector is often called speed.

• The magnitude of a vector is expressed using the same letter as

the vector but without the arrow on top of it. 

 A A Aof   Magnitude   )(

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P

Q

Initial

Point

TerminalPoint

22, y x

11, y x

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8

Some Vector Properties

• Two vectors that have thesame direction are said to beparallel.

• Two vectors that have

opposite directions are saidto be anti-parallel.

• Two vectors that have thesame length and the samedirection are said to beequal no matter where theyare located.

• The negative of a vector is avector with the samemagnitude (size) butopposite direction

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3-2 Addition of Vectors—Graphical Methods

For vectors in one

dimension, simple

addition and subtraction 

are all that is needed. You do need to be careful

about the signs, as the

figure indicates.

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Easy Adding… 

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All these planes have the same reading on

their speedometer. (plane speed not speed

with respect to the ground (actual speed)

What

factor is

affecting

theirvelocity?

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A BC

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Addition of Vectors—Graphical Methods

If the motion is in two dimensions, the situation is

somewhat more complicated.

Here, the actual travel paths are at right angles to

one another; we can find the displacement by

using the Pythagorean Theorem.

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Perpendicular VectorsWhen 2 vectors are perpendicular , you may use the

Pythagorean theorem.

95 km,E

55 km, N

 A man walks 95 km, East

then 55 km, north.

Calculate his

RESULTANT

DISPLACEMENT.

kmcc

bacbac

8.109120505595Resultant

22

22222

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ExampleA bear, searching for food wanders 35 meters east then 20 meters north.

Frustrated, he wanders another 12 meters west then 6 meters south. Calculate

the bear's displacement. 

3.31)6087.0(

6087.23

14

93.262314

1

22

Tan

Tan

m R

q 

q 

35 m, E

20 m, N

12 m, W

6 m, S

- =23 m, E

- =14 m, N

23 m, E

14 m, N

The Final Answer: 

R

3.31,93.26

3.3193.26

m

m

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To add vectors, we put the initial point of the second

vector on the terminal point of the first vector. The

resultant vector has an initial point at the initial point

of the first vector and a terminal point at the terminalpoint of the second vector (see below--better shown

than put in words).

v w

Initial point of vv Move w over keeping

the magnitude and

direction the same.

To add vectors, we pu t the in i t ial point of the second

vector on the term inal poin t of the f irst vector . The

resultant vector has an initial point at the initial point

of the first vector and a terminal point at the terminalpoint of the second vector (see below--better shown

than put in words).

To add vectors, we put the initial point of the second

vector on the terminal point of the first vector .  The

resultant vecto r has an ini t ia l po int at the ini t ia l po int

of the first vecto r and a term inal point at the term inalpo in t of the second vector  (see below--better shown

than put in words).Terminal

point of w

w

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18

Vector Addition• Vector C of a vector sum of vectors A and C.

• Example: double displacement of particle.

• Vector addition is commutative (the order of vector

addition doesn’t matter).

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The negative of a vector is just a vector going the opposite

way.

v

v

A number multiplied in front of a vector is called a scalar . It

means to take the vector and add together that many times.

v

v

vv3

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u

v

wvu   u

v

w3w   w

w

Using the vectors shown,

find the following:

vuu

vvwu 32

uu w

w w

v

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23

Vector Subtraction

• Subtract vectors:

)(   B A B A

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3 4 Addi V t b C t

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3-4 Adding Vectors by Components

Any vector can be expressed as the sum 

of two other vectors, which are called its

components. Usually the other vectors are

chosen so that they are perpendicular  to

each other.

3 4 Addi V t b C t

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3-4 Adding Vectors by Components

If the components are

perpendicular , they can be

found using trigonometric 

functions.

Remember:

soh

cah

toa

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 Adding Vectors by Components

A 

B 

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 Adding Vectors by

Components

A  B 

Transform vectors so they are head-to-

tail. 

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 Adding Vectors by

Components

A  B B

y 

Bx

Ax

Ay

Draw components of each vector... 

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 Adding Vectors by

Components

A  B 

By 

BxAx

Ay

 Add components as collinear vectors! 

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 Adding Vectors by

Components

A  B 

Combine components of answer using the head to tail

method...

Ry 

Rx 

R 

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 Analytical (component)

Method• Polar Form of Vectors

• = ,  

  =   =  

• Example

  = 10 , 45°   = 10   = 450 

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Caution

•  Addition of vectors in polar form cannot bedone algebraically

Ex. A = 5 km, 45 deg

B = 4 km, 135 deg

C = 3 km, 270 deg

R = 12 km, 450 deg

Vectors can only be added algebraically ifthey are parallel or antiparallel

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Component Form

• = ,  

• = cos + sin  

–  and  – = 10 , 45° 

– = 10 (45) + 10 sin 45  – = 7.07 + 7.07  

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  = 10 , 45° 

  = 10 (45) + 10 sin 45    = 7.07 + 7.07  

7.07

7.07 10  

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= 20 , 120° 

= 20 (120) + 20 sin 120   = 10 + 17.32  

17.32 20

10  

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• = 7.07 + 7.07  • = 10 + 17.32  

• R = -2.93 km x + 24.39 km y

17.32

10  

7.07

7.07

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Graphical Representation of the

 Analytical Method•  = 12 km, 30 deg

•  = 6 km, 60 deg

•  +  =  

  

 

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= + 

= + 

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Component Method

x y

 

D1 620km, 0 deg 620km (cos 0) 620km (sin 0)D2 440km, 315 deg 440km (cos 315) 440km (sin 315)

D3 550 km, 233 deg 550km (cos 233) 550km (sin 233)

x y

D1 620km, 0 deg 620km 0kmD2 440km, 315 deg 311km -311 km

D3 550 km, 233 deg -331 km -439 km

R 600 km -750 km

•  ++ =  

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Adding vectors:

1. Draw a diagram; add the vectors graphically.

2. Choose x  and y  axes.

3. Resolve each vector into x  and y  components.

4. Calculate each component using sines andcosines.

5. Add the components in each direction.

6. To find the magnitude of the vector, use:

= +  

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Direction:

1. Resolve what quadrantis the vector pointing at?

2. Get the Reference Angle =−

 

3. if Quadrant 1 =  Quadrant 2 =  Quadrant 3 = +  

Quadrant 4 =  

+x

+y

 –x

+y

 – x

 – y +x – y

Q1Q2

Q3 Q4

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From Component to Polar

• Magnitude of R = 600 + 750 =960  

•  Angle of R = − − =51 deg at Q4•   360-51 = 309 degrees

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= 600 750  

750

600  

Adding Vectors by

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Adding Vectors by

Components

Mail carrier’s displacement. 

A rural mail carrier leaves the post office and drives

22.0 km in a northerly direction. She then drives in a

direction 60.0 

south of east for 47.0 km. What is her

displacement from the post office?

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Magnitude  Direction 

22  90 

47  -60 

X   Y 

0 22

23.5 -40.70319398

23.5 

-18.70319398 

=

+

 

= 23.5 + 18.7  

= 23.5 + 18.7 =30.0 

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Scaling Vectors

x y

3*D1 = 1860 km, 0 deg

2*D2 = 880 km, 315 deg4*D3 = 2200 km, 233 deg

R

3 1 2 2 + 4 3 =  

D1 620km, 0 deg 3*D1 = 1860 km, 0 deg

D2 440km, 315 deg 2*D2 = 880 km, 315 degD3 550 km, 233 deg +4*D3 = 2200 km, 233 deg