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Vectors. What is the difference between a vector and a scalar number?. Vectors. Scalar. Both have a magnitude but only vectors have a direction. What kinds of things can be represent by vector. Displacement - Magnitude- how far you went Direction - which way. Velocity --

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Vectors

What is the difference between a vector and a scalar number?


Vectors

Scalar

Both have a magnitude but only vectors have a direction


What kinds of things can be represent by vector

Displacement-

Magnitude- how far you went

Direction - which way

Velocity --

Magnitude- speed

Direction - which way

Acceleration --

Magnitude- change in velocity / time

Direction - which way

Forces

Magnitude- how hard you are pushing or pulling

Direction - which way


Vectors

The direction can be indicated many different ways


If I moved 3 meters and wanted to say which way I could say in terms of common directions

Up

Right

Left

Down


North in terms of common directions

East

West

South

Or I could use compass directions...


270 in terms of common directionso

0o

180o

90o

Or I could use the angle in degrees...


+y in terms of common directions

+x

-x

-y

Or I could use Cartesian coordinates


Or I can simply use an arrow to show direction. This is called graphically representing a vector (with a picture)

When using a picture, what does the length of the arrow show?


Any way will work but generally one works the best for a given situation.

3 meters to the right

3 meters East

3 meters at 0o

Dx = +3 meters

3 meters


Just as scalars can be added given situation.

3 apples + 5 apples = 8 apples

VECTORS can be ADDED too (but watch direction)

3 + 5 doesn’t always equal 8


WHEN 2 scalar numbers are added, given situation.

the answer is called the SUM.

WHEN 2 Vectors are added,

the answer is called the RESULTANT


First we will look at adding vectors which are parallel given situation.

Are these “vectors” parallel?


8 m given situation.

If you walk 3 meters to the right and 5 meters to the right. What would your displacement be?

3 m

5 m

8 meters to the right

What is the answer called?


2 m given situation.

If you walk 3 meters to the right and 5 meters to the left. What would your displacement be?

3 m

5 m

2 meters to the left

3 + 5 = 2 ????


2 m given situation.

If you had used positive and negative x to add these how would you have done it?

- x

left

+ x

right

3 m

+ 3 m

- 5 m

5 m

- 2 m

When vectors are parallel, you simply add or subtract the numbers!!!


Vector addition also applies to Velocity given situation.

(or any other vector for that matter)


Have you ever walked on a moving walkway at an air port? given situation.

standing still on it

walking with it

walking against it

3 velocities

walkway (relative to..) you relative to walkway you relative to the ground


SCALE given situation.

= 10 km/hr

A plane is flying 60 km/hr North in still air.

60 km/hr


A wind starts blowing at 10 km/hr given situation.

North, what is the resulting

speed of the plane

SCALE

= 10 km/hr

70 km/hr

(resultant)

or

70 km/hr NORTH compared to the ground

the plane travels at

60 km/hr relative to the air

air travels 10 km/hr

relative to the ground


What if the plane did a U-turn and pointed south? given situation.

SCALE

= 10 km/hr

50 km/hr

(resultant)

or

50 km/hr SOUTH

60 km/hr

(plane)

10 km/hr

(wind)


What would the signs be on the two vectors being added ? given situation.

SCALE

= 10 km/hr

+

- 60 km/hr + 10 km/hr = -50 km/hr

-

-

60 km/hr

(plane)

10 km/hr

(wind)

+


SCALE given situation.

= 10 km/hr

-50 km/hr

(resultant)

or

50 km/hr SOUTH

- 60 km/hr + 10 km/hr = -50 km/hr

-

60 km/hr

(plane)

10 km/hr

(wind)

+


When vectors point in the given situation.same direction.

They simply add up!!

6 m

10 m

When vectors point in opposite directions,

they CANCEL (at least partially)

because they have opposite signs

10 m

6 m


A boat can move at 9 m/s in still water. If the water flows at 3 m/s. What will the boat’s velocity be if goes downstream? Upstream?


If you walked at 3 m/s. What will the boat’s velocity be if goes downstream? Upstream?10 m to the East and then 6 m to the East you would be 16 m East from where you started(your displacement)

6 m

10 m

10 m

6 m

What if you walked 10 m to the East and then 6 m to the North. Are you 16 m away from where you started?


ADDING NON-Parallel VECTORS at 3 m/s. What will the boat’s velocity be if goes downstream? Upstream?

to find the resultant


If you walk 10 m East and 6 m North, at 3 m/s. What will the boat’s velocity be if goes downstream? Upstream?

where would you be?

6 m

10 m

When 2 perpendicular vectors are added, use the parallelogram (rectangle) method.

1.) Complete the parallelogram

2.) Draw the resultant from start to finish


What is the resultant at 3 m/s. What will the boat’s velocity be if goes downstream? Upstream?

Scale 1 dm = 2.24 m

10 m

6 m

6 m

10 m


To specify the resultant we will need to say 2 things about it

10 m

6 m

6 m

10 m

Magnitude

(from the length of the line and a scale)

DIRECTION

Measuring an angle with a protractor


Find the resultant: it

SKIP

Scale: 1 dm = 5 m


Find the resultant using the parallelogram method: it

8 m West & 19 m South

Scale: 1 dm = 5 m

Draw the vectors


If the vectors are perpendicular, it

they form a right triangle.

Then we ALSO can use PYTHAGOREAN’s THEORUM to find the RESULTANT


A right triangle has a 90 degree angle. it

The side opposite it is always the longest side called the hypotenuse

Hypotenuse

Side

Side



C it

A

B

A2 + B2 = C2

This is known as the Pythagorean Theorem


Find the hypotenuse: it

3 cm

4 cm

2.5 m

10.5 m


ON the back of your vector WS it

Use the Pythagorean Theorem to find the resultant

for #2. BOX your work below


Find the resultant graphically and using the Pythagorean Theorem :

15 m due south + 25 m due west

1 dm = 5 m

SKIP

Draw the vectors


A plane plane is flying 75 km/hr due North. Theorem :

A crosswind picks up which blowing 25 km/hr due east. What is the velocity of the plane with the wind (compared to the ground)?

What happens to the speed of the plane?

Check the answer using the Pythagorean Theorem


Non-perpendicular vectors Theorem :

are added the same way


You can’t simply move an arrow on a piece of paper. It must be carefully redrawn with the same angle and length

90

90

0

0

0

0


You can’t simply move an arrow on a piece of paper. It must be carefully redrawn with the same angle and length


90 must be carefully redrawn with the same angle and length

0

0


Find the resultant: must be carefully redrawn with the same angle and length

1 dm = 6.3 m


Find the resultant must be carefully redrawn with the same angle and length

1 dm = 10 km/hr


Add the following vectors must be carefully redrawn with the same angle and length

& find the resultant

35 m heading 58o N of E

54 m heading 12o S of W

Scale 1 dm = 10 m


VECTOR WS 2 must be carefully redrawn with the same angle and length


& book problems must be carefully redrawn with the same angle and length

page 40-41

3, 4, 19, 20, 23, 24


2 Vectors can be added to make must be carefully redrawn with the same angle and length

to form their equivalent resultant

Resultant =


A must be carefully redrawn with the same angle and lengthY

AX

A single vector can be thought of as the sum of its two component vectors.

A



A component vectors.

AY

AX

A single vector can be thought of as the sum of its two component vectors.

=

+

A = AX + AY


Breaking a vector into its horizontal and vertical components is called: Vector RESOLUTION

It just involves forming a rectangle, with the given vector as the diagonal

A


One way of doing this is to draw a horizontal line touching one end and a vertical line touching the other. Where they meet it is a corner

A


Your Campsite has the displacement vector below. one end and a vertical line touching the other. Where they meet it is a corner

Find the East-West & North-South component vectors.

Scale 1 dm = 10 km

camp


A plane is flying 55 one end and a vertical line touching the other. Where they meet it is a cornero North of West at 150 km/hr.

Find the component vectors?

How fast is it moving North?

How fast is it moving West?

1 dm = 50 km / hr

Draw the vectors


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