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 wentDirection - which way

Velocity -- Magnitude- speedDirection - which way

ForcesMagnitude- how hard you are pushing or pullingDirection - which way

Acceleration -- Magnitude- change in velocity / timeDirection - 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

RightLeft

Up

Down

EastWest

North

South

Or I could use compass directions...

0o180o

270o

90o

Or I could use the angle in degrees...

+x-x

+y

-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 right3 meters East

3 meters at 0o

x = +3 meters

3 meters

Just as scalars can be added

3 apples + 5 apples = 8 apples

VECTORS can be ADDED too (but watch direction)

3 + 5 doesn’t always equal 8

WHEN 2 Vectors are added,the answer is called the RESULTANT

WHEN 2 scalar numbers are added,the answer is called the SUM.

First we will look at adding vectors which are parallel

Are these “vectors” parallel?

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

3 m 5 m

8 m

8 meters to the right

What is the answer called?

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

3 m

5 m

2 m

2 meters to the left

3 + 5 = 2 ????

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

3 m

5 m

2 m

+ 3 m

+ xright

- xleft

- 5 m

- 2 m

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

Vector addition also applies to Velocity

(or any other vector for that matter)

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

standing still on it

walking with it

walking against it

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

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

SCALE

= 10 km/hr

60 km/hr

A wind starts blowing at 10 km/hrNorth, what is the resulting speed of the plane

SCALE

= 10 km/hr

the plane travels at60 km/hr relative to the air

air travels 10 km/hr relative to the ground

70 km/hr(resultant)

or70 km/hr NORTH compared to the ground

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

SCALE

= 10 km/hr

60 km/hr(plane)

10 km/hr(wind)

50 km/hr(resultant)

or50 km/hr SOUTH

What would the signs be on the two vectors being added ?

SCALE

= 10 km/hr

60 km/hr(plane)

10 km/hr(wind)

+-

-

+

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

SCALE

= 10 km/hr

60 km/hr(plane)

10 km/hr(wind)

-50 km/hr(resultant)

or50 km/hr SOUTH

-

+

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

When vectors point in the 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

6 m

10 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 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 m10 m

6 m

10 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 VECTORSto find the resultant

If you walk 10 m East and 6 m North,where would you be?

6 m

10 m

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

1.) Complete the parallelogram2.) Draw the resultant from start to finish

What is the resultant

6 m

10 m

6 m

10 m

Scale 1 dm = 2.24 m

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

6 m

10 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: Scale: 1 dm = 5 m SKIP

Find the resultant using the parallelogram method: 8 m West & 19 m South Scale: 1 dm = 5 m

Draw the vectors

If the vectors are perpendicular, 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.

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

Side

SideHypotenuse

Which side is the hypotenuse?

CA

B

A2 + B2 = C2

This is known as the Pythagorean Theorem

3 cm

4 cm

2.5 m

10.5 m

Find the hypotenuse:

ON the back of your vector WS

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

Draw the vectors

SKIP

A plane plane is flying 75 km/hr due North. 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 vectorsare 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

0

90

0

0

90

0

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

same angle and length

0

90

0

Find the resultant: 1 dm = 6.3 m

Find the resultant1 dm = 10 km/hr

Add the following vectors & find the resultant

35 m heading 58o N of E54 m heading 12o S of W

Scale 1 dm = 10 m

VECTOR WS 2

& book problemspage 40-413, 4, 19, 20, 23, 24

2 Vectors can be added to make to form their equivalent resultant

Resultant =

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

A

AX

AY

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

A

AX

AY

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

AX

AY

A= +

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.Find the East-West & North-South component vectors.Scale 1 dm = 10 km

camp

A plane is flying 55o North of West at 150 km/hr.Find the component vectors?

1 dm = 50 km / hr

Draw the vectors

How fast is it moving North?

How fast is it moving West?