1.1 vectors

LECTURE 1

VECTORS

IB Physics Power Points

Topic 2

Kinematics

www.pedagogics.ca

Sections of Lecture 1• What is a vector?

• Describing vectors

• Multiplication and Division of vectors

• Addition and Subtraction of Vectors

• Vectors in same or opposite directions

• Vectors at right angles

• All other vectors

What is a vector?

Vectors are quantities that have magnitude (size) AND direction.

For example:

35 m is a scalar quantity.

35 m [East] is a vector quantity.

Quantities like temperature, time, mass and distance are examples of scalar quantities.

Quantities like force, velocity, and displacement are examples of vector quantities.

Describing Vectors

Consider a gravitational force of 65 N [down]

magnitude unit direction

Vectors are graphically represented by arrows:

Length = magnitude

Drawn to scale

F

When describing vectors, it is convenient to use a standard (x,y) reference frame.

+ y directionNorthup

- y directionSouthdown

- x directionWestleft

+x directionEastright

In physics, convention dictates vector direction (angle) is measured from the x axis of the frame of reference.

+ y directionNorthup

- y directionSouthdown

- x directionWestleft

+x directionEastright

30o

S

[right, 30o above horizontal][30o N of E][E, 30o upwards]

In math it is common to describe direction differently:

For example:

To a physicist, the direction of this vector is [60o S of W]

OR . . . measure the angle of direction in a counter clockwise direction from East. The same vector direction is now described as [240o] This method has advantages.

Multiplication and Division with Vectors

Any vector can be multiplied or divided by a scalar (regular) number.

The multiplication or division will change the magnitude of the vector quantity but not the direction.

For example:

F 2F

Addition and Subtraction of Vectors

Part 1: Parallel vectors

Example 1: Blog walks 35 m [E], rests for 20 s and then walks 25 m [E]. What is Blog’s overall displacement? (what is displacement??)

Solve graphically by drawing a scale diagram.

1 cm = 10 m

Place vectors head to tail and measure resultant vector.

1s

2s

60 m [E]s

Solve algebraically by adding the two magnitudes. WE CAN ONLY DO THIS BECAUSE THE VECTORS ARE IN THE SAME DIRECTION.

SO 35 m [E] + 25 m [E] = 60 m [E]

Solve Graphically

1 2x x

1x

2x

10 m [E]x

Example 2: Blog walks 35 m [E], rests for 20 s and then walks 25 m [W]. What is Blog’s overall displacement?

resultant

Algebraic solution, we can still add the two magnitudes. WE CAN ONLY DO THIS BECAUSE THE VECTORS ARE PARALLEL!

WE MUST MAKE ONE VECTOR NEGATIVE TO INDICATE OPPOSITE DIRECTION.

SO 35 m [E] + 25 m [W]

= 35 m [E] + – 25 m [E]

= 10 m [E]

Note that 25 m [W] is the same as – 25 m [E]

Addition and Subtraction of Vectors

Part 2: Perpendicular VectorsExample 3: Blog walks 30 m [N], rests for 20 s and then walks 40 m [E]. What is Blog’s overall displacement?

Solve Graphically

1 2x x 1x

2x

length 50 m 37o

ox 50 m [37 N of E]

Addition and Subtraction of Vectors

Part 2: Perpendicular VectorsAlgebraic solution: use trigonometry.

Diagram does not need to be to scale.

2 230 40 50 mx

1x

2x

ox 50 m [37 N of E]

1 1 30tan tan 3740oy

x

Part 3 - Adding multiple vectors (method of components)

Consider the following 3 displacement vectors:A student walks

3 m [45o N of E]

6 m [N]

5 m [30o N of W]

Vectors are illustrated here to scale.

To determine the resultant displacement, add the individual vectors graphically by drawing them head to tail.

The resultant displacement is 11 m [102o] OR 11 m [78o N of W]

Show scale diagram solution here

+ x directionWestleft

It is not possible to add the vectors shown in the diagram by the algebra methods discussed previously. The vectors are neither parallel or perpendicular.

Method of Components

To add vectors that are not in the same or perpendicular directions – use method of components.

All vectors can be described in terms of two components called the x component and the y component.

This is a displacement vector of magnitude 36 m

The vector has been placed on an x,y coordinate axis with the tail at the origin (0,0)

Method of Components

The x component of this vector is shown by the green line.

The y component of this vector is shown by the pink line.

x

y

Method of Components

You may have noticed that the original vector is just the sum of the two vector components

xy+

36 m [34o N of E]There is NO difference in displacement between walking 36 m [34o N of E] and walking x m [E] followed by y m [N]

Now consider this vector

18 m [45o S of E]

36 m [34o N of E]

Graphically addedto a second vector

Resultant

We can’t use algebra to add these vectors directly BUT we could use algebra to add their components.

18 m [45o S of E]

x1

y1

36 m [34o N of E]

To get the same resultant

x2

y2

x1 x2

y1

y2

Conclusion: Adding the vectors graphically using their components produces the same result.

BONUS: Components can be added using math methods because all x components are in the same plane as are all y components. Furthermore, x and y components are perpendicular and can be added to each other using Pythagorean theorem.

It really is elementary!

Now you need to meet my old

Greek friend

Meet Hipparchus, considered to be the father of trigonometry.

He is here to remind you of our fictitious Mayan hero Chief Soh Cah Toa

adjacent

hypotenuse

opp

osit

e

Determining Components

The x component = S (cos )

The y component = S (sin )

x

y

Ssin

cos

ySxS

Where S is the magnitude of the original vector and

is the angle between the original vector and the x axis

Examples:Resolve the following vectors into x and y components

Vector X Y

15.2 m [27o N of E]    

12.7 ms-1 [56o]  

45.0 N [48o N of W]    

725 m [205o]    -657 m -306 m

7.1 ms-1 10.5 ms-

1

13.5 m 6.90 m

-30.1 N 33.4 N

Solving vector problems

Example: Blog starts his walk at the old oak tree. He walks 55 m [42o S of E] to Point A. He then walks 75 m [185o] to Point B. He then walks a final 62 m [78o N of W] to Point C. What is Blog’s overall displacement?

Step 1 (always)

Sketch a diagram

(does not have to be to scale but it helps) AB

C

Solving vector problemsStep 2: Resolve vectors into x and y components and add them.

Vector X Y

55 m [42o S of E]  40.9 -36.8 

75 m [185o] -74.7  -6.54

62 m [78o N of W]  -12.9 60.6 

55cos(42) 75cos(185) 62cos(78)40.9 74.7 12.946.7

x x x x

x

x

x

s A B Csss

Solving vector problemsStep 2: Resolve vectors into x and y components and add them.

Vector X Y

55 m [42o S of E]  40.9 -36.8 

75 m [185o] -74.7  -6.54

62 m [78o N of W]  -12.9 60.6 

55sin(42) 75sin(185) 62sin(78)36.8 6.54 60.6

17.3

y y y y

y

y

y

s A B Csss

Solving vector problemsStep 3: Use sum of components to determine resultant.

Sx = -46.7 m OR 46.7 m [W]

Sy = 17.3 m or 17.3 m [N]

Sy = 17.3 m

Sx = - 46.7 m

Use trig to find length and direction of resultant.

49.8 m [20.3o N of W]