Vectors

Ms. Arra C. Quitaneg

What do we measure to

tell how hot or how cold

an object is?

What describes how

much space is occupied

by an object?

What is used to

describe how

massive an object is?

Temperature

Volume

Mass

A physical quantity which is described by a

single number with a unit

Examples: time, temperature,mass, density

electric charge

Calculations with scalar quantities use

operations of ordinary arithmetic .

A physical quantity which has both

magnitude and direction.

When handwritten, use an arrow:

When printed, will be in bold print with an

arrow:

A

A

When dealing with just the magnitude of a

vector in print, an italic letter will be used:

A or | |

The magnitude of the vector has physical

units

The magnitude of a vector is always a

positive number

A

Displacement

Force

Velocity

Acceleration

A vector is represented by an arrow

Arrowhead points to the direction.

Length of the arrow represents the

magnitude of the vector.

Vector’s magnitude can also be expressed in

numbers.

The magnitude of a vector is always positive.

Direction of the vector can be represented

by a sign ( + or - ) depending on the sign

convention ex: North +, South –

Direction is also expressed in words.

Ex. South, Southwest

Angle is also used to describe a vector’s

direction.

ex. 300 South of East, + 500 , + 1200 , - 2200

Scale: 1 cm = 1 m

d= 4m , East

Scale: 1 cm = 1 m

d= 4m , 45 0 North of East

A = 4 m, 50 0 N of E

A =4 m, 400 E of N

Parallel vectors- if two vectors have the same

direction

Equal vectors- if two vectors have the same

magnitude and direction

Negative of a vector- a vector having the same

magnitude as the original but has opposite

directions.

Antiparallel – when two vectors have opposite

directions, whether their magnitudes are

equal or not.

Scale: I cm= 1 unit

4 Newtons East

5 m, North

10 m/s, 20 0 N of E

8 m, 40 0 E of N

7 N, 40 0 W of S

6 m, 50 0 S of E

12 m, 40 0 S of W

5m, + 230 0

8 m/s, - 1800

Result of vector addition is called as vector

sum or resultant vector.

R = (A +B) + C = A + (B +C)

It obeys the associative law, order of addition

makes no difference.

R = A + B = B + A

It obeys commutative law, order of terms

in a vector sum does not matter.

Head to tail method

1. From the origin, the vector, based

on its direction and magnitude is

drawn.

2. Then, the tail of the second

vector is attached to the head of

the first.

3. Resultant vector is drawn from the

tail of the first vector to the head

of the last vector.

When vectors are parallel, just add magnitudes and keep the direction.

Ex: 50 m/s east + 40 m/s east = 90 m/s east

When vectors are antiparallel, just subtract the smaller magnitude from the larger and use the direction of the larger.

Ex: 50 m/s east + 40 m/s west = 10 mph east

R

θ

Use a protractor to measure the angle.

A = 5 N, East

B = 7 N, 60 deg N of E

Find the resultant

of A and B. A = 11 N @ 35° N of E

A 35° N of E

B = 18 N @ 20° N of W

B

20° N of W

R

57° N of W

R = 14.8 N

@ 57° NW

Parallelogram method

1. Plot the given vectors from the origin

( based on their magnitude and directions

2. Reflection of the vectors are drawn, until a

parallelogram is formed.

3. Resultant vector is the diagonal of the

parallelogram. Measure the length and the

angle of the resultant vector.

VECTOR ADDITION EXERCISES:

1.Erwin walks 300 East and stops to rest

and then continues 400 m East. What is

his total displacement?

2.Mimi walks home from school 300 m East

and remembers that she has to bring

home her Science book which a classmate

borrowed. She walks back 500m West to

her classmate’s house. What is the

resultant displacement of Mimi?

3. Early in the morning, Patrick would

always jog 500 m East from his house

and turns North and walks 300 m. What

is his displacement from his house?

4. An ant crawls on a tabletop. It moves 2

cm East and turns 3 cm 40 0 North of

East and finally moves 2.5 cm North.

What is the ant’s total displacement?

The analytical

method of vector

addition!!!

Magnitude of the resultant vector is given by the arithmetic sum of the magnitudes of the individual vectors.

Direction is unchanged.

Magnitude of the resultant vector is given

by the arithmetic difference of the

magnitudes of individual vectors.

Direction of the resultant vector is the

same as the direction of a vector with

greater magnitude.

Use Pythagorean theorem

22

yx AA A

The Pythagorean theorem

Trigonometric functions are used in

solving for the directions.

sin = opp/ hyp

cosine = adjacent/ hyp

tan = opp/ adjacent

SOH CAH TOA

x

y

A

A1tan

Component method-

1. Draw each vector.

2. Find the x and y components of

each vector ( Vector resolution).

Use trigonometric functions.

3. Find the sum of x – components

and y components.

4. Use pythagorean theorem to get

the resultant vector.

The river was moving with a velocity of 3

m/s, North and the motor boat was

moving with a velocity of 4 m/s, East.

What would be the resultant velocity of

the motor boat (i.e., the velocity relative

to an observer on the shore)?

A plane can travel with a speed of 80 mi/hr

with respect to the air. Determine the

resultant velocity of the plane (magnitude

only) if it encounters a

a. 10 mi/hr headwind.

b. 10 mi/hr tailwind.

c. 10 mi/hr crosswind.

d. 60 mi/hr crosswind.

A motor boat traveling 5 m/s, East

encounters a current traveling 2.5 m/s,

North.What is the resultant velocity of the

motor boat?

What are the x and y components a force of

500 N exerted at an angle 300 from the

horizontal.

A man walked 5m, 400 North of East. What

are the x and y components of his

displacement?

A man walked 2m, 250 North of East, then 5m

300 West of North. What is the man’s

displacement? (magnitude and direction)

An airplane with a velocity of 120

km/h,heads North. However wind blows with

a speed of 30 km/h at an angle of 400 North

of West. What is the velocity of the airplane

relative to the ground?

An airplane has a speed of 285 km/h with respect to

the air. There is a wind blowing at 95 km/h at 30 deg

north of east with respect to Earth. In which

direction should the plane head in order to land at

an airport due north of its present location. What

would be the plane’s speed with respect to the

ground?

You are piloting a small plane and you want to reach

an airport 450 km due south in 3 hours. A wind is

blowing from the west at 50 km/h. What heading

and airspeed should you choose to reach your

destination in time?

An airplane,moving at 375 m/s relative to the

ground, fires a cannon shell at a speed of 782 m/s

relative to the plane. What is the speed of the shell

relative to the ground.

1. Solve for the x and y components of each given

vector. Indicate the sign of each component.

2. Add all x components, add all y components.

3. Use pythagorean theorem to get the magnitude

of the resultant vector.

4. Solve for the direction using tan theta function.

5. Express your answers in 2 decimal places.