Vectors & Scalars

Physics 11

Vectors & Scalars•A vector has magnitude as well as direction.

•Examples: displacement, velocity, acceleration, force, momentum

•A scalar has only magnitude

•Examples: time, mass, temperature

Vector Addition – One Dimension

A person walks 8 km East and then 6 km East.

Displacement = 14 km East

A person walks 8 km East and then 6 km West.

Displacement = 2 km

Vector Addition

21 DDDR

22

21 DDDR

Example 1: A person walks 10 km East and 5.0 km North

kmkmkmDR 2.11)0.5()0.10( 22

RD

D2sin

0121 5.26)2.11

0.5(sin)(sin

km

km

D

D

R

Order doesn’t matter

Graphical Method of Vector AdditionTail to Tip Method

1V

2V

3V

RV

Graphical Method of Vector Addition“Head-to-Tail” Method

1V

2V

3V

RV

1V

2V

3V

Graphical Method of Vector Addition Parallelogram Method

Helpful hints about parallelograms:•All four angles add to equal 360o

•Opposite angles are equal

Properties of Parallel Lines

Subtraction of Vectors

• Negative of vector has same magnitude but points in the opposite direction.

• For subtraction, we add the negative vector.

Multiplication by a Scalar

• A vector V can be multiplied by a scalar c• The result is a vector cV that has the same direction but a

magnitude cV• If c is negative, the resultant vector points in the opposite

direction.

Adding Vectors by Components• Any vector can be expressed as the sum of two other

vectors, which are called its components (i.e. Vx & Vy).

• Components are chosen so that they are perpendicular to each other.

Trigonometry Review

Opposite

Adjacent

Hypotenuse

Hypotenuse

Oppositesin

Hypotenuse

Adjacentcos

cos

sin

Adjacent

Oppositetan

Pythagorean Theorem:(Hypotenuse)2 = (Opposite)2 + (Adjacent)2

Adding Vectors by Components

If the components are perpendicular, they can be found using trigonometric functions.

sinVVy

cosVVx

V

Vy

V

Vx

cos

sin

Adj

Opptan

Hypotenuse

Oppositesin

Hypotenuse

Adjacentcos

Adding Vectors by Components

• The components are effectively one-dimensional, so they can be added arithmetically:

Signs of Componentsy

x

y

x

R

R

y

x

R

R

y

x

R

R

y

x

R

R

Adding Vectors by Components

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:

V

Vysin

Relative Velocity

•Relative velocity considers how observations made in different reference frames are related to each other.

Example: A person walks toward the front of a train at 5 km/h (VPT). The train is moving 80 km/h with respect to the ground (VTG). What is the person’s velocity with respect to the ground (VPG)?

TGPTPG VVV

hkmhkmhkmVPG /85/80/5

Relative Velocity

•Boat is aimed upstream so that it will move directly across.

•Boat is aimed directly across, so it will land at a point downstream.

•Can expect similar problems with airplanes.

Practise Problems• #1, page 70

• #9, page 71

• #12, page 71

• #40, page 73

• #41, page 74