1

Adding VectorsVectors are ‘magnitudes’(ie: values) with a direction

The speed and direction of a car is a vector.

The strength and direction that you push something with a force is a vector.

Your ‘displacement’ if you walk some city blocks is a vector.

Vectors can be added.

If you take Physics you will need to be rather proficient with vectors!

2

Two Displacement Vectors

Vector A: Walking three blocks north

Vector B: Walking four blocks east.

A

B

‘Tail’ of the vector

‘Head’ of the vector

3

4

3

Adding Vectors: Tail to Head Method

Put the ‘tail’ of Vector B to the ‘head’ of Vector A.

A B

BAC

Vector C is the result of adding Vector A to Vector B. Vector C is called the ‘resultant’.

Vector C has a length and a direction.

3

4

4

Show adding again

A B

B

B

B

B

BAC

3

4

So the resultant Vector C is found by adding the tail of Vector BB to the head of Vector AA. Vector C has a length and a direction; but what is that length and direction?

5

0

1020

30405060708090

100

110

120

130

140150

180

160170

Find Length and Direction of Resultant

A B

BAC

3

4Just measure the length of C with a ruler and its angle with a protractor

0

1

2

3

4

5

The resultant Vector C is 5 long in a direction 53 degrees to the right of Vector A. We have measured the length and the direction of Vector C.

6

Resultant Velocity: Swimming‘Velocity’ is a speed in a given direction

Sheena is swimming partly ‘upstream’ and across a river at 4 km/h in a direction 45° from the shore. The current is at 2 km/h and is parallel to the river of course.

Find the actual speed and direction that Sheena is actually moving by adding the two vectors.

Current

4 cm for 2 km/h

Sheena

8 cm

for 4

km

/h

Shoreline

Scale: A length of 2 cm is 1 km/h45°

Current

4 cm for 2 km/h

6 cm

for

3 k

m/h

Measure the length of the resultant and its angle. Sheena is swimming at 3 km/h at an angle of 28 degrees to the left of her swimming velocity

28°

Or you could say Sheena’s direction is 45° + 28° = 73° from the shore

7

Vectors: Using More ToolsThe previous examples were conceptual; they gave you the idea.

Let us be a bit more rigorous in our method of adding vectors.

It works out better if we are drawing vectors if we use a ‘grid’ and a certain ‘scale’ of measurement.

The grid will help us measure the angles more easily; the scale will allow us to make vectors the right length for different situations.

And it will help if we agree on a better way to measure angles. We need an angle that is the ‘zero’ angle, and then measure everything from that.

A ‘grid’:

The bearing navigation reference for measuring angles:

[000°]

[090°]

[180°]

[270°]

[135°]

North

EastWest

South

8

Vectors: Motion ExampleHave you ever noticed boat or airplane motion?

The boat or airplane will be pointing (‘heading’) in a certain direction with a certain speed, but actually traveling in a different direction (its ‘course’) and speed because the air itself is moving (moving air is called ‘wind’!)

windwindno

wind

The direction the aircraft is pointing is called its ‘heading’ and its speed through the air is called ‘airspeed’. Its actual speed relative to the ground is called its ‘ground speed’; its actual direction is called its ‘track’.

It is the same idea with boats in a current

9

Aircraft Velocity ExampleAn aircraft is flying North [000°] at 200 mph. The wind is from the west [270°] at 80 mph. What is the actual resultant motion vector of the airplane?

1. Establish a scale. Lets say that one cm is 20 mph.

Scale:Scale:1 cm = 20 mph1 cm = 20 mph

2. Draw the aircraft vector from some convenient point. Call it vector A

[000°]

[90°]

[180°]

[270°]

[000°]

[90°]

[180°]

[270°]

A

3. Add the wind vector to the head of the aircraft vector. Call it W.10

cm

for

200

mph

10 c

m f

or 2

00 m

ph

W

4cm= 80mph

4. Measure the length and direction of the resultant Vector R.

The resultant is 10.8 cm long, so 216 mph

180170160150140

130120110100

90807060504030

02010 The angle

is [022°]

The aircraft is really moving in a direction [022°] at 216 mph.

R

10

Displacement Vectors: Water BalloonTerrance and Monique are playing with water balloons. They are initially standing together. Monique runs 8 metres from Terrance in direction [045°] then 6 metres in direction [120°]. How far must Terrance throw his water balloon and in which direction to get Monique?

Add Monique’s two displacements together by adding her two vectors.

Scale: 1 cm = 1 metre

1cm = 1m

8 mete

rs@

[045

°]6 meters@[120°]

11.2 cm or 11.2

meters@[076°]

[000°]

[90°]

[180°]

[270°]

[000°]

[90°]

[180°]

[270°]

180

170160

150140

130

120

110

100

90

80

7060

50

40

30

0

2010

180

170160

150140

130

120

110

100

90

80

7060

50

40

30

0

2010

Terrance must throw his balloon in the direction [076°] and a distance of 11.2 meters to bean Monique

Displacement just means ‘change in position’

11

Properties of Vectors: Commutative Law

You know that 2 + 4 = 6 and 4 + 2 = 6. This is called the commutative law of addition. The order in which you add numbers does not matter.

You also know that the commutative law works with matrices

54

12

81

03

81

03

54

12

The commutative law works for vectors too!

ABBA

Going two blocks east and three blocks south is the same as going three blocks south and two blocks east!

12

Commutative Law of Vectors

ABBA

A

+B

=A

B

BA

B

+ A

B A

AB

=

13

Subtracting VectorsCan you subtract vectors?Of course!

Just add an ‘opposite’ vector! Just like 5 – 3 = 5 + (–3)

)( BABA

A B

B

A

B

BA

B

BA

You will not subtract Vectors in Grade 12 Math, but you will in Physics!

14

Multiply Vectors by a ScalarHere is Vector A

A

What is 2*Vector A? It is really just Vector A + Vector A (that is what multiplying by two is!)

A

A+

A

A

A

*2

15

Multiply a Vector by a VectorCan you multiply a Vector by a Vector?Yes, you can!

But that is more of a university lesson, so we will not do it here in Grade 12!

XYZ

When you multiply Vector X by Vector Y you end up leaving the paper in a third dimension; so save that idea for university!

YXZ

*

16

Adding Vectors: Parallelogram Method

There is another way to add vectors graphically!

It is a bit more intuitive for some people.

The parallelogram method!

All you do is transfer the directions of the vectors to the head of each vector and where they cross is the resultant

17

Parallelogram Method ExampleRick and CJ are trying to pull out a tree stump. Rick is pulling with a force of 100 lbs in the direction N45W, CJ with a force of 160 lbs N45E. What is the resultant force on the tree stump?

20 lbs = 1 cm

Just transfer the direction of each vector to the head of the other vector

Measure 9cm or 180 lbs in a direction 34° to the left of CJ or N11W

RickCJ34°

100 160

Res

ulta

nt f

orce

18

Vector Addition Using Trigonometry

So do you walk around with graph paper and a ruler and a protractor?

Even I do not do that!

So lets figure out how to do vectors using trigonometry, at least you only need a calculator for that!

Have you noticed that adding two vectors has actually been using a triangle?

19

Adding Vectors using Trigonometry

8

6

Caution! Lengths are not to scale now, we can just use trig, so we don’t have to accurately draw vectors. We can just sketch them

From Pythagoras we know thatR2 = A2 + B2A

B

So R2 = 82 + 62

So R is 10 long if you do the calculationR

But what is the angle ?

OppositeA

dja

cent

9.36)8/6(tan 1

8

6tan

adj

opp

20

Cosine and Sine Law

The last example had a 90° corner. But not all triangles and vectors have a 90° corner. Pythagoras does not work if there is no 90 ° angle!

So we will need to use our Cosine and Sine laws from Grade 10 if the triangle is not a right triangle!

Abccba cos2222

C

c

B

b

A

a

sinsinsin

And you will need to remember some other basic rules from Grade 10 Geometry!

a

bc

A

B C

21

[000°]

[90°]

[180°]

[270°]

[000°]

[90°]

[180°]

[270°]

Vectors using Sine and Cosine Law

An aircraft is flying on a heading of [045°] at 300 mph through the air. The wind is fromfrom a direction of [300°] at 60 mph. What is the actual vector motion of the aircraft? (track and ground speed)

A

W

R

Caution, we are just sketching our vectors here. The directions and lengths may not be to scale or accurate!

300

60

[000°][045°]

[120°]

75°

105°

)105cos()60)(300(260300 222 RmphR 8.320

105sin

8.320

sin

60

Therefore: = 10.4°. So R is in direction [045 °] + 10.4 ° = [055.4 °]

The aircraft is flying at 321 mph in a direction [055°]

22

Example 2: Sine and Cosine Law

Nathan is out snowmobiling and gets lost. He knew he went 6 km north [000°] then he turned 60° to the right and went another 4 km. How far is he from home and in which direction?

6

460°

[000°]

[90°]

[180°]

[270°]

[000°]

[90°]

[180°]

[270°]

120cos)4)(6(246 222 R762 R So R = 8.7 km

Cosine Law:

120° Sine Law:

8.7

km

sin

4

120sin

7.8

398.07.8

120sin4sin

5.23)398.0(sin 1

So Nathan is 8.7 km from home on a bearing of [023.5°]