Vector

Vectors and Scalars

A vector has magnitude as

well as direction.

Some vector quantities:

displacement, velocity, force,

momentum

A scalar has only a magnitude.

Some scalar quantities: mass,

time, temperature

Distance: A Scalar Quantity

A scalar quantity:

Contains magnitude only and consists of a number and a unit.

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

A

B

Distance is the length of the actual path taken by an object.

s = 20 m

Displacement—A Vector Quantity

A vector quantity:

Contains magnitude AND direction, a number, unit & angle.

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

A

B D = 12 m, 20o

• Displacement is the straight-line separation of two points in a specified direction.

q

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

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.

Applications of Vectors VECTOR ADDITION – If 2 similar vectors point in the SAME

direction, add them.

• Example: A man walks 54.5 meters east, then another 30 meters east. Calculate his displacement relative to where he started?

54.5 m, E 30 m, E +

84.5 m, E

Notice that the

SIZE of the arrow

conveys

MAGNITUDE and

the way it was

drawn conveys

DIRECTION.

Applications of Vectors VECTOR SUBTRACTION - If 2 vectors are going in

opposite directions, you SUBTRACT.

• Example: A man walks 54.5 meters east, then walks

30 meters west. Calculate his displacement relative

to where he started?

54.5 m, E

30 m, W +

24.5 m, E

Easy Adding…

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

their

velocity?

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.

Perpendicular Vectors When 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.

kmc

c

bacbac

8.10912050

5595Resultant 22

22222

Example A 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

mR

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 q

3.31,93.26

3.3193.26

m

m

Addition of Vectors—Graphical Methods

Even if the vectors are not at right

angles, they can be added graphically by

using the tail-to-tip method.

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 terminal

point of the second vector (see below--better shown

than put in words).

v w

Initial point of v v

w

Move w over keeping

the magnitude and

direction the same.

Terminal

point of w

w

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.

vv

vv3

u

vw

vu u

vw3

ww

w

Using the vectors shown,

find the following:

vu u

vvwu 32

uu w

ww

v

3-3 Subtraction of Vectors, and

Multiplication of a Vector by a Scalar

In order to subtract vectors, we

define the negative of a vector, which

has the same magnitude but points

in the opposite direction.

Then we add the negative vector.

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.


If the components are

perpendicular, they can be

found using trigonometric

functions.

Remember:

soh

cah

toa

Adding Vectors by Components

The components are effectively one-dimensional,

so they can be added arithmetically.


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 and cosines.

5. Add the components in each direction.

6. To find the length and direction of the vector, use:

and .

3-4 Adding Vectors by

Components

Example 3-2: 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?

3-4 Adding Vectors by

Components

Example 3-3: Three short trips.

An airplane trip involves three legs,

with two stopovers. The first leg is

due east for 620 km; the second leg

is 45° south of east (315) for 440

km; and the third leg is at 53°

south of west (233°), for 550 km, as

shown. What is the plane’s total

displacement?

Analytical Method

• Polar Form of Vectors

• 𝐴 = 𝐴 , 𝜃𝐴

– 𝐴 = 𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑣𝑒𝑐𝑡𝑜𝑟 𝐴

– 𝜃𝐴 = 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑣𝑒𝑐𝑡𝑜𝑟 𝐴

• Example

– 𝐴 = 10 𝑘𝑚, 450

– 𝐴 = 10 𝑘𝑚

– 𝜃𝐴 = 450

Caution

• Addition of vectors in polar form cannot be done 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 if they are parallel or antiparallel

Component Form

• 𝐴 = 𝐴 , 𝜃𝐴

• 𝐴 = 𝐴 cos 𝜃𝐴 𝑥 + 𝐴 sin 𝜃𝐴 𝑦

– 𝑦 and 𝑥 𝑎𝑟𝑒 𝑐𝑎𝑙𝑙𝑒𝑑 𝑢𝑛𝑖𝑡 𝑣𝑒𝑐𝑡𝑜𝑟𝑠

– 𝐴 = 10 𝑘𝑚, 450

– 𝐴 = 10 𝑘𝑚𝑐𝑜𝑠(45)𝑥 + 10𝑘𝑚 sin 45 𝑦

– 𝐴 = 7.07 𝑘𝑚 𝑥 + 7.07 𝑘𝑚 𝑦

𝐴 = 10 𝑘𝑚, 450

𝐴 = 10 𝑘𝑚𝑐𝑜𝑠(45)𝑥 + 10𝑘𝑚 sin 45 𝑦

𝐴 = 7.07 𝑘𝑚 𝑥 + 7.07 𝑘𝑚 𝑦

7.07 𝑘𝑚

7.07 𝑘𝑚 10 𝑘𝑚

– 𝐵 = 20 𝑘𝑚, 1200

– 𝐵 = 20 𝑘𝑚𝑐𝑜𝑠(120)𝑥 + 20𝑘𝑚 sin 120 𝑦

– 𝐵 = −10 𝑘𝑚 𝑥 + 17.32 𝑘𝑚 𝑦

17.32 𝑘𝑚

20 𝑘𝑚

−10 𝑘𝑚

• 𝐴 = 7.07 𝑘𝑚 𝑥 + 7.07 𝑘𝑚 𝑦

• 𝐵 = −10 𝑘𝑚 𝑥 + 17.32 𝑘𝑚 𝑦

• R = -2.93 km x + 24.39 km y

17.32 𝑘𝑚

−10 𝑘𝑚

7.07 𝑘𝑚

7.07 𝑘𝑚

Graphical Representation of the

Analytical Method

• 𝐴 = 12 km, 30 deg

• 𝐵 = 6 km, 60 deg

• 𝐴 + 𝐵 = 𝐶

𝐴

𝐵

𝐴

𝐵

𝐴𝑥

𝐴𝑦

𝐵𝑥

𝐵𝑦

𝑐

𝑐𝑥 = 𝑎𝑥 +𝑏𝑥

𝑐𝑦 = 𝑎𝑦 +𝑏𝑦

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 0km

D2 440km, 315 deg 311km -311 km

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

R 600 km -750 km

• 𝐷1 +𝐷2+ 𝐷3 = 𝑅

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 and cosines.

5. Add the components in each direction.

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

𝑅 = 𝑥2 + 𝑦2

Direction:

1. Resolve what quadrant is the vector pointing at?

2. Get the Reference Angle 𝜽𝒓𝒆𝒇 = 𝒕𝒂𝒏−𝟏𝒚

𝒙

3. if Quadrant 1 𝜽 = 𝜽𝒓𝒆𝒇

Quadrant 2 𝜽 = 𝟏𝟖𝟎 − 𝜽𝒓𝒆𝒇

Quadrant 3 𝜽 = 𝟏𝟖𝟎 + 𝜽𝒓𝒆𝒇

Quadrant 4 𝜽 = 𝟑𝟔𝟎 − 𝜽𝒓𝒆𝒇

+x

+y

–x

+y

– x

– y +x

– y

Q1 Q2

Q3 Q4

From Component to Polar

• Magnitude of R = 6002 + −750 2 =960 𝑘𝑚

• Angle of R = 𝑡𝑎𝑛−1−750

600=51 deg at Q4

• 360-51 = 309 degrees

𝑅 = 600 𝑘𝑚 𝑥 −750𝑘𝑚𝑦

−750 𝑘𝑚

600 𝑘𝑚

Scaling Vectors

x y

3*D1 = 1860 km, 0 deg

−2*D2 = − 880 km, 315 deg

4*D3 = 2200 km, 233 deg

R

3𝐷1 − 2𝐷2 + 4𝐷3 = 𝐻

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

D2 440km, 315 deg −2*D2 = − 880 km, 315 deg

D3 550 km, 233 deg +4*D3 = 2200 km, 233 deg

Problem Set

• 𝐴 = 4 𝑘𝑚, 30

• 𝐵 = 13 𝑘𝑚, 120

• 𝐶 = 10 𝑘𝑚, 240

• 𝐷 = 7 𝑘𝑚, 310 • Find the magnitude and direction of the following

1. 𝐴 + 𝐵 + 𝐶 + 𝐷 = 𝐸

2. 𝐴 − 𝐵 + 𝐶 + 𝐷 = 𝐹

3. 𝐴 + 𝐵 − 𝐶 + 𝐷 = 𝐺

4. 𝐴 + 𝐵 + 𝐶 − 𝐷 = 𝐻

5. 𝐴 + 𝐵 − 𝐶 − 𝐷 = 𝐽

6. 𝐴 − 𝐵 − 𝐶 + 𝐷 = 𝐾

Vector

Documents

Transcript of Vector