Math, Trigonometry and Vectors1. Geometry

1. Trigonometry2. Trig Definitions3. Inverse functions4. Pythagorean Theorem

2. Scalars and Vectors3. Properties of vectors4. Unit vectors

1. Function of Unit vectors2. Dot Product3. Cross Product

5. Other math1. Graphs & Plots

GeometryQuick Question 1

33º

What is the angle equal to?

Trigonometry

θ This picture represents the origins of the trigonometric relations. Inscribe a right

triangle inside a circle. Now, the angle theta is one of the angles of the triangle. The ratios of the various sides make up the trigfunctions.

α

a) α = 27∘

b) α = 57∘

c) α = 33∘

d) α = 90∘

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Trig Definitions

Here's a familiar image.

To make predictive models of the physical world, we'll need to make visual models, which we can then turn into analytical statements.Many of the models will be geometric in nature. Thus, we'll need things like the trigonometric relations to establish relations betweenthe components. In the triangle shown here, one angle is marked with a . The sides are labeled in relation to this angle: opposite,adjacent, and hypotenuse. So, is defined as the ratio of the side labeled opposite to the side known as the hypotenuse.

.

.

.

a

bc

This one might be a little less familiar, but the same rules apply.

Quick Question 2

Which of the following statements is (are) true?

θ

sin(θ)

sin(θ) =opphyp

cos(θ) =adjhyp

tan(θ) =oppadj

a) sin(α) = cos(β)b) δ = β

c) β+ γ = ϵ

d) tan(α) = tan(γ)

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Inverse functionsWe can also use the inverse trigonmetric functions.

a

b

c

The inverse trig functions take the ratio of lengths (a dimensionless number) and return an angle (in degrees or radians).

It's 450 meters from the corner of Hamilton Place and Broadway to 142nd St and Broadway. It's 489 meters from the same cornerto 142nd if you walk along Hamilton Place. a) What is the angle between Hamilton and Broadway?

Pythagorean Theorem

( ) = θsin−1opphyp

( ) = θcos−1adjhyp

( ) = θtan−1oppadj

Example Problem#1:

θ = ( ) =cos−1450489

23.04∘

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a

b

c

We'll use this relationship allthetime.

. This is also known as Euclid's 47th proposition from the first book of the Elements.

It's 450 meters from the corner of Hamilton Place and Broadway to 142nd St and Broadway. It's 489 meters from the same cornerto 142nd if you walk along Hamilton Place. b) How far is it from Broadway to Hamilton Pl. walking along 142nd St?

map

Scalars and VectorsThese are two different mathematical or physical entities.

Scalars: A scalar quantity is completely specified by a single value with an appropriate unit and has no direction. (e.g. $20)

Vectors: A vector quantity is completely described by a number and appropriate units plus a direction. (e.g. person walks 2 km E)

Some Examples:Scalars: Temperature, Speed, Distance, length, density

Vectors: Displacement, Velocity, Force, Weight

When thinking out the physical world, you should intuitively notice that certain quantities or phenomena have different effects

+ =a2 b2 c2

Example Problem#2:

b = = = 191.4 m−c2 a2− −−−−−√ −4892 4502

− −−−−−−−−√

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depending on which way they are pointed, in other words, their effects depend on their direction. For example, it's a lot easier to walkwith the wind blowing in the same direction as your motion, rather than the other way: walking against the wind. There are two vectorquantities at play in this example. Your direction of motion (that would be inferred from your velocity vector) and the velocity of thewind. When they point in the same direction, your motion is aided by the wind, when they are in opposite directions, your motion isimpeded. Not only do the magnitudes of these two quantities matter, but so do their directions. And thus, we need to use vectors.

Vector vs. Scalar exampleA particle travels from A to B along the path shown by the dotted red line. This is the distancetraveled and is a scalar

The displacement (change in position) is the solid line from (a) to (b). The displacement is independent of the path taken between thetwo points displacement is a vector (it has length and direction).

a

b

This image illustrates the difference between displacement and distance traveled. Looking at the dotted line, which represents thedistance traveled, compared to the solid displacement vector, we can see that a) its magnitude is probably much larger than thedisplacement vector, and b) it doesn't have a clear direction associated with it.

NotationWhen writing math by hand, just put an arrow on top of the variable. This will indicate it is a vector:

In printed text, you'll see vectors either in bold face: or with an arrow:

If we want to refer to the magnitude only of a vector quantity, we can use absolute value bars: , or just in italics: .

Wind Map

Properties of vectorsTwo vectors are equal if they have the same magnitude and the same direction

if and they point along parallel lines

All of the vectors shown are equal in magnitude and direction, thus they are equal.

x

y

It might be helpful to think of vectors a notation like this: . Thinking this way we can see that the definition of thevector only requires two elements, the magnitude and the direction. If we can another vector , with the same magnitude anddirection, it would necessarily have to be equal to .

Addition of VectorsAdding two scalar quantities is easy. We just add them like we would add any normal quantity.

However, vectors involve more math. We have to also take into account which way they are pointing.

A⃗

A A⃗

|A| A

A = B |A| = |B|

A = (mag, dir)B

A

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Imagine we walk along two displacement vectors and . What is the resultant displacement? Or, what is ?

Vector Addition: GraphicallyTo add

1. Arrange the vectors tip to tail.2. Connect the tip of to the origin of .

Link to Vector Addition Sim.

Quick Question 3

Which of the following is the resultant for ?

Example Problem#3:

A⃗ B⃗ +A⃗ B⃗

+A⃗ B⃗

A⃗ B⃗

+P ⃗ Q⃗

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Negative of a VectorThe negative of a vector is simply a vector with the same magnitude, but pointed in the opposite direction.

The resultant of

Vector SubtractionTo Subtract two vectors, say , all we need to do is add the negative of to .

Since,

Quick Question 4

Which of the following is the resultant for ?

Quick Question 5

Here is a vector . Which if the four vectors below it (A - D) should I add to to make vector ?

+ (− ) = 0A⃗ A⃗

−A⃗ B⃗ B⃗ A⃗

− = + (− )A⃗ B⃗ A⃗ B⃗

−P ⃗ Q⃗

PP Q

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P

A B C D

Q

Multiplication of a Vector times a ScalarThe result of the multiplication or division by a scalar is a vector.

x

y

x

y

x

y

The magnitude of the vector is multiplied or divided by the scalar.

Vector ComponentsThe components of a vector are the parts of a vector that point along a given axis. We'll use the Cartesian Coordinate System mostoften.

Here we see the and components of the vector .

We can see that the component, , points all along the axis, while the component, , points only along the axis.

x

y

x y A

x Ax x y Ay y

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x

y

Here's another vector decomposed into its and components.

Quick Question 6

Here are the components of :

Which diagram represents ?

x

y

2

2

-2

-2

-4

-4

4

4x

y

2

2

-2

-2

-4

-4

4

4

x

y

2

2

-2

-2

-4

-4

4

4x

y

2

2

-2

-2

-4

-4

4

4

We can use these vector components to add two arbitrary vectors together. (notice that and are not at right angles to each other.)

x

y

x

y

We'll combine the components of and to get the components of .

A x y

R

= +4, = +3Rx Ry

R

A B

A B C

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Once we have the components of , we can use the pythagorean theorem to get the magnitude of .

x (km)

y (km)

N

A car travels 20 km due N and then 35 km in a direction 60º W of N. Find the magnitude and direction of the car’s resultantdisplacement.

Other mathGraphs & PlotsVisual Information is Powerful.

Functional Relations

y

x0

Constant

y

x0

Linear

= +Cx Ax Bx

= +Cy Ay By

C C

|C| = +C2x C2

y

− −−−−−−√Example Problem#4:

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y

x0

Quadratic

Quick Question 7

Which graph would best represent a plot of the function:

x

y

0

0x

y

0

0x

y

0

0x

y

0

0

A B C D

Quick Question 8

Which graph would best represent a plot of the function:

x

y

0

0x

y

0

0x

y

0

0x

y

0

0

A B C D

Quick Question 9

At what point is the slope of this function equal to zero?

y = 2x

y = x14

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x

y

0

0

A B C D

Quick Question 10

At what point is the slope of this function the most negative?

x

y

0

0

A B C D

Quick Question 11

h

g

0

0Which of the following functions could describe this plot? ( is a constant)

Quick Question 12

Here is a series of temperature measurements for two patients. Both areclearly getting colder. Based only on this data, who should you treat first?

a

a) g(h) = a+ h

b) g(h) = −h

c) g(h) = a− h

d) g(h) = h

e) g(h) = h−1

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MeasurementTime(min)

PatientATemp(°F)

PatientBTemp(°F)

1 98.6 98.6

2 98.4 98.55

3 98.2 98.4

4 98 98.15

Simultaneous SolutionsVery often, we'll have two equations and two unknown variables. There are several algebraic tactics that can be employed to solve the'system'.

Quick Question 13

Which combination of these 3 vectors has the largest magnitude?

Quick Question 14

What is the sum of vectors ?

a+ 4b

a− 4==24b

(1)(2)

a) + +A⃗ B⃗ C⃗

b) + −B⃗ A⃗ C⃗

c) − +A⃗ B⃗ C⃗

d) + +C⃗ B⃗ A⃗

A + B+ C + D

a) Rb) −Rc) Gd) −Ge) 0

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A

B

RG

C

D

Quick Question 15

You walk 100 miles east from point A to point B. How far did you walk intotal?

Quick Question 16

You walk 100 miles east from point A to point B. How far are you from yourstarting point?

a) 0 milesb) 100 milesc) 200 miles

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Quick Question 17

You walk 100 miles east from point A to point B, then 100 miles north frompoint B to point C. How far did you walk in total?

Quick Question 18

a) 0 milesb) 100 milesc) 200 miles

a) 0 milesb) less than 200 milesc) 200 miles

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You walk 100 miles east from point A to point B, then 100 miles north frompoint B to point C. How far are you from your starting point?

a) 0 milesb) less than 200 milesc) 200 miles

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