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MATH 40 APPLIED VECTORS NOTES INTRODUCTION Vectors are just ‘quantities with direction’ You use vectors all the time and probably don’t realize it Important in Physics too DEMONSTRATION You start at point A You move to Point B Then you move to Point C How far did you move to get to B? _______ What direction did you go to get to B? _______ How far did you move from B to C? _______ What direction did you go from B to C? _______ How far did you end up moving from A to C and in what direction? __________________

B

C

A Scale: 1 ‘unit’ per square

VECTORS AND SCALARS Each of your straight movements is a vector. They have a length and a direction. Length is also called ‘Magnitude’ A ‘Vector’ has a magnitude and a direction! List some things that have only a magnitude (scalars):

Measures that only have a magnitude are called ‘scalars’

‘Speed’ is a scalar

MA40SA_Vectors_Notes.doc

Revd: 28 Feb 2007

2 Now list some things that have a magnitude and a direction

EQUAL Vectors You start at point A You move to Point B Ray starts at C and moves to D

B

D

How far did you move to get to B? _______ What direction did you go to get to B? _______ How far did Ray move from C to D? _______ What direction did Ray go from C to D? _______ We call these two vectors: _______________

A

C

Scale: 100 m per square Or 1 cm = 250 meters

VECTOR NOTATION

r A . Notice the arrow over the variable. In this case the We show vectors normally like this: r Vector is vector A . Or we can show the start point and end points like this: AB EQUAL VECTORS Two vectors are equal if they have the same magnitude and direction. Think of the Snowbirds in flight! Each and everyone of them has the same vector most of the time. If their vectors were not the same they would be moving apart, or worse, moving together

3 PARALLEL VECTORS Two vectors are parallel if they go in the same direction even if their magnitudes are r r different. A and B are parallel . All equal vectors are necessarily parallel. You draw and label two parallel vectors!

r A

Parallel r B

Parallel and co-linear r A

OPPOSITE VECTORS

r B

Two Vectors are opposite if they have the same magnitude but opposite directions. Measure the two of these in magnitude, and confirm they are parallel and opposite. r r A and B are opposite . You draw and label two opposite vectors. MULTIPLY A VECTOR BY A SCALAR r You go 5 blocks north. Your vector is A r Rays goes twice as far. His vector is B

r r B = 2* A

We say that So Ray’s vector is 10 blocks north! r r You plot a vector that is C = 3 * A

r B

r A

ADDING VECTORS (USING GEOMETRY) The Tail-to-Head method is a simple way to add many vectors that are displacements.

B

You walk from A to B. Draw a vector to show it. You then walk from B to C. Draw a vector to show that

C

Now make a dashed arrow from A to C. Don’t forget the arrowhead. It shows what your resultant vector was. A resultant is the result of combining two vectors What is the magnitude and direction of the each of the vectors? (Measure with a ruler in mm and protractor for degrees!)

A

Resultant − −− >

− −− >

− −− >

AB =

BC =

AC =

4 MEASURING DIRECTION There are several ways of measuring direction. Direction is usually measured relative to something. Often the reference is ‘north’ of a map or the ‘vertical’ on a page. Sometimes a direction is just relative to one of the vectors (eg:15° to the right of Vector A)

North 0°

Navigation (Bearing) Method The most common method North is at 0° and usually points to the top of a page or a map The other angles look like at the right

330°

30°

300°

60°

West 270°

East 90°

Notice the angle is measure clockwise from North 120°

240°

You draw a vectors that is going in the direction [30°]

210°

You draw a vector that is going in the direction [245°]

150°

South 180°

DIRECTION – DEGREE- DIRECTION METHOD Used in many physics books W60N

More intuitive for some people. Angles are given relative to the main ‘quadrantal’ directions of N,E,S,W as per diagram at the right. Example: N30E means ‘Go 30° East of North’ or Go North then 30° East of that Not very good for computers because of ambiguity: two names for same angle Example ambiguity: S30W = W60S You draw a vector that has direction S10E You draw a vector that has direction N30W

N30E

W30N W

N60E

W

E W30S E45S S30E

S30W

S

5 REVIEW OF PLOTTING OF VECTORS Plot and label the following vectors a.

r A ; 10 meters at [030°]

b.

r B ; 12 meters at [120°]

c.

r C ; 8 meters N30E

d.

r D ; 12 Meters S45W

e.

r r A + B

f.

r 2* C

g.

r r C + D

ADDING VECTORS – GEOMETRY – THE PARALLELOGRAM METHOD

Scale: 1 square = 1 meter or 1cm = 2.5 m

A litre of milk weighs very close to 10 Newtons

Also called the ‘tail-to-tail’ method. Useful when vectors have a common start point Especially used for forces acting on a common point or object! Terry is pulling on a tree stump with a force of 1000N at E30N. Tara is pulling on the tree stump with a force of 800N at N30E. What is the force that acts on the tree stump? Construct a parallelogram out of the two vectors having a corner at the common point of the vectors. The resultant vector sum is the diagonal of the parallelogram. The result is 1610N at [048°] or 1610N at N52E or 1610N at 18° to the right of Tara’s force.

Scale: 1 square = 100 Newtons or 1 cm = 12.5 Newtons (N)

The triangle method of tail- to-head still works for these problems too!

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earth

PRACTICE PARALLELOGRAM METHOD An asteroid is being acted on by the Sun and Jupiter’s gravity!

Sun Pull

The sun pulls at 10GigaNewtons to the North. Jupiter pulls at 8 GigaNewtons to the East. Does it look like the asteroid will hit our earth?? Jupiter Pull

asteroid

Scale: 1 square = 1GigaNewton MATHEMATICAL METHODS OF SOLVING VECTORS The above geometry methods work well, but they rely on us being very accurate in drawing and having good graph paper and protractors and rulers. And of course, a computer cannot possibly use these methods to solve vector problems. We need a way to solve problems with just mathematics: algebra, and trigonometry. There are actually many ways to work with vectors, we will only look at two here. THE COMPONENT METHOD DEMONSTRATION 120

r Tyler goes A :16 miles at a direction of r [030°]. He then goes B :[120°] for 12 miles. What was Tyler’s resultant vector?

12 16

How far right did he go on vector A?

How far up did he go on vector A?

30°°

How far right did he go on vector B? Scale: 1 square = 1 mile How far up did he go on vector B?

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Add the rights (x), and ups (y) separately. That is a resultant displacement vector from the start point. What is the magnitude and direction? THE COMPONENT METHOD STEPS Do a rough sketch of the problem, it doesn’t have to be accurate since you will not be using geometry to figure it out. Sketch in the x and y axis Find the x component of each vector using trig Find the y component of each vector using trig Add the x components and the y components together to get the relative resultant point (x, y)

r Use the distance formula to calculate the magnitude of the resultant vector V . d = (V x ) 2 + (V y ) 2

(like Pythagoras and the Grade 10 distance formula)

Use the trigonometry ‘Tan’ function to calculate the angle of the resultant relative to the xaxis. (watch the negative signs, don’t lose them!!!) Vy tan(θ ) = Vx Express the magnitude and direction of the resultant vector (usually relative to north) EXAMPLE COMPONENT METHOD

r r r Find C = A + B given: r A = 12 miles at [045°] r B = 10 miles at [150°]

Rough Sketch Ax = 8.48 y Ay = 8.48

Bx = 5

r A

Ax = 12 * sin(45) = 8.48 right A y = 12 * sin( 45) = 8.48 up B x = 10 * sin(30) = 5 right B y = 10 * cos(30) = 8.66 down Therefore: Cx = 8.48 + 5 = 13.48 right Cy = 8.48 – 8.66 = –0.18 up Find Magnitude of Vector C

By = –8.66

r B

θ

length = (13.48) 2 + ( −0.18) 2 = 13.48 units

r C

x

8 Find Direction of Vector C C y 0.18 tan(θ ) = = = −0.0134 C x 13.48 −1

therefore θ = tan (0.0134) = 1°

PRACTICE PROBLEM 1 COMPONENT METHOD OF ADDING VECTORS Use the component method to solve the following. Make a sketch at the right. An aircraft is heading in a direction of 360° (relative to north) at 300 km/h. The wind is from 270 degrees at 80 km/h. What is the speed and direction (that is: the velocity!) of the resultant aircraft motion?

Notice the wind direction is given as from the direction it blows! All weather reports are like this!

PRACTICE PROBLEM 2 COMPONENT METHOD OF ADDING VECTORS Use the component method to solve the following. Make a sketch at the right. Two boys are lifting a rock. One pulls with a force of 200N at an angle of 70° to the ground. The other pulls from the other side at an angle of 45° to the ground with a force of 300N.

So the resultant is 1 degree below the x axis, or in other words 91 degrees from north.

r

Therefore: C =13.48 units at [091°]

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SOLVING VECTOR TRIANGLES WITH THE SINE AND COSINE LAW The Sine Law and Cosine Law are very powerful trigonometry. They allow you to discover all parts of a triangle even if it is not a right angle triangle. If you intend on going to University for any science or Engineering you should get my special package on the Sine and Cosine Law. A very quick summary of the Sine and Cosine Law is given below but you will not have to use it in any Grade 12 work for this Vectors course. B a

c A

b

C

The Sine Law

The Cosine Law

Basically says that the big side is opposite the big angle and that the sin of the angle/the length of the side opposite is constant for any given triangle

a 2 = b 2 + c 2 − 2bc * cos( A)

sin A sin B sin C = = a b c

OTHER NOTES: In navigation problems there are special terms you may encounter. Aviation

Air Speed is the speed of the aircraft through the air. Heading is the direction the aircraft is pointing. Ground Speed is the resultant magnitude of the speed along the ground after adding the wind vector to the speed and heading of the aircraft. Track or course is the subsequent direction of the vector that results. Thus the Heading and Airspeed Vector is added to the Wind Vector to get the Course and Groundspeed The angle between the heading of a vessel and its course is called the drift.

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Heading and Air Speed

Wind

Drift

Course and Ground Speed

Caution. The normal way to express a wind direction is the direction from which it blows. So the wind in this example might be 310 degrees at 30 km/h if north is to the top

Forces The measure of force is called the Newton. On the earth’s surface a mass of 1 kg has a force downwards of 10 Newtons (N). (Since gravity on earth has a value of 10). Force vectors are shorter on the moon where the effect of gravity is much smaller, so a given mass exerts less force downwards on the moon. On the moon the weight of a 1 kg mass would only be 2 Newtons. Force is achieved in a wire or rope through tension. Thus a rope pulling with 100N of force can also be said to have 100N of tension. The force exactly opposite another force to keep the forces balanced is called the equilibrant. In other words, a body presses down with 10 Newtons and something else (an opposite force vector) has to hold it up with 10 Newtons; that is the ground or some ropes, etc. Mathematical Operations - Subtracting Subtracting a vector from another is the same as multiplying the one by a negative one to get the opposite vector and then adding. Subtracting is just adding an opposite, just like in pure numbers. We don’t worry about subtracting vectors in Grade 12.

11 DETAILED EXAMPLE OF ADDING VECTOS BY COMPONENTS

Components of a Vector This point is 8 to y

1

n 0u

the right and 6 above the start point of the vector

its

A y= 6 The y-component of Vector A, Ay, is 6

36.9°

A x= 8

The x-component of Vector A, Ax, is 8

x Any vectors can be ‘resolved’ into an ‘x-component’, Ax and a ‘y-component’, Ay

Components by Trigonometry SO/H CA/H TO/A s nit nuse u 10 ypote H

Opposite

y

36.9° Adjacent o

Adj

Ay= 10 sin (36.9°) Ay = 6

Ax

36.9 ) = = Ax=cos( 10*cos(36.9°) =8 Hyp 10

x Any vectors can be ‘resolved’ into an ‘x-component’, Ax and a ‘y-component’, Ay using trigonometry

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Without Graph Paper, Just Trig!! Resultant Component: Cx = Ax + Bx = 6 – 10 = – 4 Resultant Component: Cy = Ay + By= 3 + 5 = + 8

r r r A+ B = C

y C =8 11. y 2 By = 11.2 * sin (26.6) By = 5

Ay = 6.7*sin(26.6) Ay = 3 6 .7

26.6° Bx = 11.2*cos(26.6) Cx = –4 Bx = 10 but –10 sincex left

26.6° Ax = 6.7*cos(26.6) Ax = 6

Add Vectors by adding their components using trigonometry

Find the resultant given its The vector endpoint ends at –4, 8 from where it started

N 8. 9

Cy = 8

By Pythagoras the length of C is given by: C2 = Cx2 + Cy2 So C2 = 16 + 64 = 80 So C = 8.9 And an angle θ is given by:

63.4° θ

Cx = –4

[296.6 °]

θ = Tan-1 (8/4) = 63.4° from the left horizontal Or you could say [296.6 °] from North

13 GLOSSARY Much of this glossary is adapted from: The U of Regina Middle Years site: http://mathcentral.uregina.ca/RR/glossary/middle/; and The Cut-the-Knot.Org Math website: http://www.cut-the-knot.org/glossary/atop.shtml See also: http://www.shodor.org/interactivate/dictionary/index.html.

Angle of depression

The angle measured below the horizon or the floor. Angle of depression

angle of elevation or elevation angle. bearing

The angle above the horizon or the floor. Angle of elevation the 3-digit angle, measured in a clockwise direction, between the north line and a given direction

14 component

In vectors, the part of a vector along a certain direction. The x component is how much a vector goes along the x direction and the y component is how much a vector goes along the y direction. r Example: given the vector V of 10 meters at [060°] from north y

Vx = 10*cos(30)= 8.6m meters Vy = 10*sin(30)= 5 meters x

The x component, Vx, is 8.6 m, the y component , Vy, is 5 meters. cosine

for an acute angle ∠A in a right triangle, the ratio of the length of the side adjacent to the angle to the length of the hypotenuse

15 Cosine Law

a trigonometric law used to solve triangles that are not necessarily right triangles

equal vectors

have the same magnitude and direction

head-to-tail

two vectors are drawn head-to-tail if the second vector begins where the first vector ends The longest side of a right angle triangle. It is always opposite the 90° angle.

hypotenuse

16 Mathematical The representation of mathematics in a simple language understood by all. notation The symbology of how we represent mathematical ideas. Example Sin (angle) = length of opposite side / length of hypotenuse. They didn’t have to call it Sin, they could have called it ‘dfahdkajh’. But we all agreed over centuries that the word SINE meant something and it had meaning. Pretty amazing since they didn’t even have books back then! opposite vectors parallel lines

have the same magnitude but act in opposite directions lines in the same plane that do not intersect

Lines m and n are parallel with a transversal. If two lines are parallel and cut by a transversal, then the following will be true: The alternate interior angles will have equal measures (congruent). ∠3 =∠6 and ∠4 =∠5 The corresponding angles will have equal measures (congruent). ∠l =∠5 and ∠2 =∠6 and ∠3 =∠7 and ∠4 =∠8 The same-side interior angles add up to 180 °. (supplementary). ∠3 + ∠5 =180 °. and ∠4 + ∠6 =180 °. If anyone of these conditions is true then the lines must be parallel. Lines that are parallel will have the same slope. parallelogram a method for finding the sum of two vectors arranged tail-to-tail method of vector addition

17 ratio

a comparison of two or more quantities

scalar quantities

Example: 2 cars is to 8 hubcaps as 4 cars is to 16 hub caps are both equivalent ratios 2 4 = 8 16 quantities that can be described by specifying their magnitude only. Example: temperature, age, …

similar triangle

Two triangles are the similar when they have all similar (or same) angles. The sides can be different lengths as long as they are proportional (in the same ratio) as the other triangle.

The big triangle and the inscribed little triangle are ‘similar’. All their angles are the same, just the sides are different lengths. The sides are also proportional: so if the bottom of the big is twice a long as the bottom of the little, the height and hypotenuse must also be twice as long.

18 Sine Law

a trigonometric law used to solve triangles

Square

The square of a number is that number multiplied by itself. The square of x would be x * x which mathematicians write as x2 Example 32 = 9 The square is an inverse function of the square root (they just ‘undo’ each other). Example:

Square root

A number that when multiplied by itself gives a desired number. If the desired number is 9, the square root is 3, since 3*3=9 In math we say: 9 = 3 . The square root is an inverse function of the square. (they just ‘undo’ each other). Example

Vector

2

3 =3

32 = 3

A directed segment representing a quantity that has both magnitude, (or length), and direction.

19 TRIGONOMETRY FORMULAS AND TABLES Reminder: SOH CAH TOA

Trig for Right Angle triangles hypotenuse

Sin(θ) = Opposite side / Hypotenuse

opposite

Cos(θ) = Adjacent side/ Hypotenuse

θ Tan(θ) = Opposite side/ Adjacent side adjacent

Make sure your calculator is in the right mode for measuring angles! Inverse Trig functions Just doing the table backwards! What is the angle, θ, formed by the two given sides at the right? 6 θ = tan −1   = 36.87 o 8

6

θ 8

Trigonometric Ratios to 3 Significant Figures Angle [Degrees]

Sin

Cos

Tan

Angle [Degrees]

Sin

Cos

Tan

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.000 0.017 0.035 0.052 0.070 0.087 0.105 0.122 0.139 0.156 0.174 0.191 0.208 0.225 0.242 0.259

1.000 1.000 0.999 0.999 0.998 0.996 0.995 0.993 0.990 0.988 0.985 0.982 0.978 0.974 0.970 0.966

0.000 0.017 0.035 0.052 0.070 0.087 0.105 0.123 0.141 0.158 0.176 0.194 0.213 0.231 0.249 0.268

46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61

0.719 0.731 0.743 0.755 0.766 0.777 0.788 0.799 0.809 0.819 0.829 0.839 0.848 0.857 0.866 0.875

0.695 0.682 0.669 0.656 0.643 0.629 0.616 0.602 0.588 0.574 0.559 0.545 0.530 0.515 0.500 0.485

1.04 1.07 1.11 1.15 1.19 1.23 1.28 1.33 1.38 1.43 1.48 1.54 1.60 1.66 1.73 1.80

20 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

0.276 0.292 0.309 0.326 0.342 0.358 0.375 0.391 0.407 0.423 0.438 0.454 0.469 0.485 0.500 0.515 0.530 0.545 0.559 0.574 0.588 0.602 0.616 0.629 0.643 0.656 0.669 0.682 0.695 0.707

0.961 0.956 0.951 0.946 0.940 0.934 0.927 0.921 0.914 0.906 0.899 0.891 0.883 0.875 0.866 0.857 0.848 0.839 0.829 0.819 0.809 0.799 0.788 0.777 0.766 0.755 0.743 0.731 0.719 0.707

0.287 0.306 0.325 0.344 0.364 0.384 0.404 0.424 0.445 0.466 0.488 0.510 0.532 0.554 0.577 0.601 0.625 0.649 0.675 0.700 0.727 0.754 0.781 0.810 0.839 0.869 0.900 0.933 0.966 1.000

62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

0.883 0.891 0.899 0.906 0.914 0.921 0.927 0.934 0.940 0.946 0.951 0.956 0.961 0.966 0.970 0.974 0.978 0.982 0.985 0.988 0.990 0.993 0.995 0.996 0.998 0.999 0.999 1.000 1.000

0.469 0.454 0.438 0.423 0.407 0.391 0.375 0.358 0.342 0.326 0.309 0.292 0.276 0.259 0.242 0.225 0.208 0.191 0.174 0.156 0.139 0.122 0.105 0.087 0.070 0.052 0.035 0.017 0.000

1.88 1.96 2.05 2.14 2.25 2.36 2.48 2.61 2.75 2.90 3.08 3.27 3.49 3.73 4.01 4.33 4.70 5.14 5.67 6.31 7.12 8.14 9.51 11.4 14.3 19.1 28.6 57.3 Infinity