Physics 20: 2.1 Vector Directions

Summary

This educational video script introduces the concept of vectors in physics, focusing on understanding direction systems. It explains the Cartesian coordinate system's directional approach starting from zero degrees and the navigation system using north, east, south, and west. The script emphasizes the importance of identifying vector directions from verbal descriptions to solve problems involving trigonometry and angle calculations. It also clarifies the difference between 'north 30 degrees east' and 'east 30 degrees north,' illustrating how to convert between various directional notations for consistency in problem-solving.

Takeaways

• 📚 The lecture begins with an introduction to vectors and the importance of understanding direction in physics.
• 🧭 The main strategy for dealing with vector questions involves breaking vectors into their horizontal and vertical components using trigonometry.
• 📐 Trigonometry will be used to solve for angles and missing sides, with sine and cosine being fundamental to finding the components of a vector.
• 📏 The Cartesian coordinate system is used for direction in physics, with angles measured from the positive x-axis (0 degrees).
• 🌐 The navigation system uses cardinal directions (north, east, south, west) to specify vector directions, such as 'North 30 degrees east'.
• 🔍 Locating the direction of a vector is crucial, as questions often describe directions in words rather than providing triangles.
• 📊 When a vector is given in terms of degrees, the reference angle is used to simplify the trigonometric calculations.
• 🔄 Understanding the complementary angles is important, as they can be used to convert between different directional descriptions.
• 🌡 The navigation system can describe directions in a variety of ways, such as 'North 45 East' or 'East 45 North', which are equivalent.
• 📍 The script emphasizes the need to be able to interpret and draw vector directions from verbal descriptions to solve physics problems.
• 📘 The lecture also mentions that different materials might use different conventions for writing angles, such as '40 degrees north of West'.

Q & A

What is the main strategy for solving vector problems in physics as described in the transcript?

The main strategy for solving vector problems in physics, as described in the transcript, is to break the vector into its horizontal and vertical components. This simplifies the process of solving for angles and missing sides using trigonometry.

How does the transcript suggest breaking down a vector like a speed of 10 meters per second at 30 degrees?

The transcript suggests using trigonometry to break down the vector. For a speed of 10 meters per second at 30 degrees, you would use sine and cosine functions to find the vertical (y) and horizontal (x) components, respectively. Specifically, y = 10 * sin(30) and x = 10 * cos(30).

What is the Cartesian coordinate system used for in the context of the transcript?

In the context of the transcript, the Cartesian coordinate system is used to determine the direction of a vector. It starts with 0 degrees on the right side, then goes up to 90 degrees at the top, 180 degrees on the left, and 270 degrees at the bottom.

How does the transcript describe the process of locating a vector direction in physics?

The transcript describes the process of locating a vector direction by starting from the zero degree mark and measuring the angle from there. For example, a vector going 10 meters at 70 degrees would be drawn starting from the origin and measuring 70 degrees upwards.

What is the reference angle mentioned in the transcript, and how is it used?

The reference angle mentioned in the transcript is the smallest angle between the terminal side of an angle and the x-axis. It is used to determine the direction of a vector when the angle is greater than 90 degrees or less than 270 degrees. For example, a 350-degree angle has a reference angle of 10 degrees.

How does the navigation system for directions differ from the Cartesian system discussed in the transcript?

The navigation system for directions, as discussed in the transcript, uses cardinal directions like north, east, south, and west instead of the mathematical degrees used in the Cartesian system. It involves determining the direction of a vector based on angles from these cardinal points.

What does the transcript mean by 'North 30 degrees east' in the context of the navigation system?

In the context of the navigation system, 'North 30 degrees east' means that the vector is directed towards the north first and then moves 30 degrees from the north towards the east to determine its exact location.

How can you convert a direction like 'South 65 degrees west' into its corresponding angle in the Cartesian system?

To convert 'South 65 degrees west' into its corresponding angle in the Cartesian system, you would draw the vector starting from the south and then measure 65 degrees towards the west from that starting point.

What is the significance of the complementary angle in the context of the transcript?

The significance of the complementary angle in the context of the transcript is to help convert between different directional systems. For example, if you have 'North 30 degrees west' and you need 'West 30 degrees north', you can use the complementary angle (60 degrees) to find the equivalent direction.

How does the transcript explain the difference between 'North 30 degrees west' and 'West 30 degrees north'?

The transcript explains that 'North 30 degrees west' means moving from the north towards the west by 30 degrees, while 'West 30 degrees north' means moving from the west towards the north by 30 degrees. These are not the same direction, and the difference lies in the starting point and the direction of the angle.

What is the reference to '40 degrees north of West' in the transcript, and how does it relate to the navigation system?

The reference to '40 degrees north of West' in the transcript is an alternative way of expressing directions, where you start from the west and then move 40 degrees towards the north. It is related to the navigation system and is equivalent to 'West 40 degrees north'.

Outlines

00:00

📚 Introduction to Vectors and Direction Systems

This paragraph introduces the concept of vectors in physics, emphasizing the importance of understanding direction systems before delving into vector calculations. The speaker explains that vectors can represent various physical quantities like displacement or speed, and gives an example of a vector with a magnitude of 10 meters per second at an angle of 30 degrees. The main strategy discussed is breaking down vectors into their horizontal and vertical components to simplify problem-solving. The paragraph also touches on the basics of trigonometry that will be used, such as sine and cosine, and the challenge of visualizing and drawing triangles based on word descriptions rather than given geometric figures. The Cartesian coordinate system is introduced as the first method for direction, starting with zero degrees on the right and moving counterclockwise to 90, 180, and 270 degrees.

05:00

🧭 Understanding Cartesian and Navigation Direction Systems

The second paragraph delves deeper into the two primary direction systems used in physics: the Cartesian system and the navigation system. The Cartesian system is familiar to those who have studied trigonometry, where directions start from the zero degree mark on the right and move counterclockwise. The navigation system, akin to GPS, uses cardinal directions (north, east, south, west) to specify the direction of a vector. The paragraph provides examples of how to interpret and draw vectors described in these systems, such as 'North 30 degrees east' or 'South 65 degrees west'. It also explains how to convert between different directional descriptions, like 'North 30 degrees west' and 'West 30 degrees north', by using complementary angles. The speaker clarifies that certain directional descriptions can be interchangeable, such as 'north 45 East' or 'east 45 North', but others are distinct and require careful interpretation. The paragraph concludes by noting that different materials may present angles in various ways, such as '40 degrees north of West', which should be understood as equivalent to the navigation system's description.

10:02

🔍 Comparing and Equating Directional Descriptions

This paragraph continues the discussion on direction systems, focusing on equating different ways of describing the same direction. The speaker illustrates how various directional phrases can point to the same location, such as '60 degrees west of North', 'West 30 degrees north', and '150 degrees in the math Cartesian system', emphasizing that they all represent the same vector direction. The paragraph also addresses potential confusion with terms like 'north of West' and 'south of East', explaining that these can be rearranged or converted to understand the direction better. For instance, '35 degrees south of East' can be understood as 'East 35 South'. The speaker encourages students to be adept at visualizing and drawing any directional scenario, regardless of how it is described, to ensure they can accurately determine the vector's direction.

Keywords

💡Vectors

Vectors are quantities that have both magnitude and direction, which are fundamental in physics for describing motion and forces. In the video, vectors are introduced as essential components for understanding directionality in physics, with examples such as displacement and speed. The script discusses how to break down vectors into their horizontal and vertical components, which is crucial for solving problems involving angles and missing sides.

💡Direction System

A direction system is a method of specifying the orientation of a vector or object in space. The video clarifies how direction systems are used in physics to locate and understand the orientation of vectors. It distinguishes between the Cartesian system used in mathematics and the navigation system used in physics and GPS, which involves directions like north, east, south, and west.

💡Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. In the context of the video, trigonometry is used to break vectors into their components by utilizing sine and cosine functions. For instance, the script mentions using sine 30 degrees to find the vertical component (y) and cosine 30 degrees for the horizontal component (x) of a vector.

💡Displacement

Displacement is a vector quantity that refers to the change in position of an object. It is mentioned in the script as an example of a vector that students will encounter. Displacement is important because it includes information about both the distance and direction of an object's movement from its initial position.

💡Speed

Speed is a scalar quantity that indicates how fast an object is moving, without regard to its direction. The script uses '10 meters per second' as an example of speed, which is then combined with a direction (such as 30 degrees) to form a velocity vector. Speed is a component of velocity when direction is considered.

💡Horizontal and Vertical Components

In the script, horizontal and vertical components refer to the two perpendicular parts into which a vector can be decomposed. This decomposition simplifies the process of solving for unknowns in physics problems. For example, a vector with a certain speed at an angle can be broken down into its x (horizontal) and y (vertical) components to make calculations easier.

💡Cartesian System

The Cartesian system is a mathematical framework used to describe positions in a two-dimensional plane using coordinates. In the video, it is contrasted with the navigation system used in physics. The Cartesian system starts with 0 degrees on the right and progresses clockwise to 90, 180, and 270 degrees, which helps in determining the direction of vectors.

💡Navigation System

The navigation system, as discussed in the script, is a method of specifying direction using cardinal points like north, east, south, and west. It is commonly used in physics and GPS. The script explains how to interpret directions such as 'North 30 degrees east' by drawing triangles to determine the vector's orientation.

💡Reference Angle

A reference angle is the angle formed between the terminal side of an angle greater than 90 degrees and the x-axis (or the north in the Cartesian coordinate system). In the script, reference angles are used to simplify the understanding of directions in both the Cartesian and navigation systems. For example, a 350-degree angle has a reference angle of 10 degrees.

💡Complementary Angle

Complementary angles are two angles that add up to 90 degrees. The script explains how to use complementary angles to convert between different directional expressions, such as changing 'North 30 degrees west' to 'West 60 degrees north'. This concept helps in understanding equivalent directional descriptions.

Highlights

Introduction to vectors in physics and the importance of understanding direction systems.

The necessity of locating a vector's direction accurately in physics problems.

Breaking down vectors into horizontal and vertical components for easier problem-solving.

Incorporation of trigonometry in vector analysis with simple sine and cosine applications.

Understanding the Cartesian coordinate system as it applies to physics.

The concept of zero degrees starting on the right side in the Cartesian system.

Locating vectors based on angles in the Cartesian system, such as 70 degrees.

The process of determining vector locations in terms of degrees, like 220 degrees.

Using reference angles in the Cartesian system, such as 350 degrees equating to 10 degrees.

The navigation system in physics, which uses north, east, south, and west for direction.

Interpreting navigational vectors, such as 'North 30 degrees east'.

Understanding the difference between 'South 65 degrees west' and other directional vectors.

The concept of complementary angles in navigational directions, like 'North 30 degrees west'.

Identifying equivalent directional vectors, such as 'North 45 East' and 'East 45 North'.

The distinction between 'North 30 degrees West' and 'West 30 degrees North'.

Dealing with different directional notations like '40 degrees north of West'.

Recognizing equivalent directional scenarios in various systems, such as '60 degrees west of North'.

Transcripts

okay so we're our next units on vectors

we're going to start with vectors but

before we get to looking at Vector

questions and doing some trigonometry

and solving for angles and missing sides

and stuff like that we have to be clear

on how our Direction system works for

physics so the first thing that you got

to be careful of is when you see

something with a direction you have to

be able to locate where that exactly

would be so most of the questions we're

going to get this for this unit is

you're going to get a some sort of

vector so it could be a displacement or

a speed or whatever so let's just

suppose it was 10 meters per second

and then they're going to tell you

that's going to go at some sort of angle

so let's suppose

it was at 30 degrees

what we're going to want to do is be

able to

break that Vector into horizontal and

vertical components then we could

basically solve the questions a lot

easier so that's going to be our main

strategy so we're going to be getting

into the trigonometry and stuff later on

so basically the as hard as the Trig's

ever going to get is

you would get sine 30

would equal in this case the red let's

call it X we'd have or let's call it y

it doesn't matter which one's which

so let's for now just keep it sort of

simple let's call that our y direction

so we'd have y divided by 10 and then on

the bottom we'd have cosine of 30.

equals adjacent over hypotenuse so X

over 10. so that's what is complicated

as the trigonometry is going to get for

this stuff which is good everybody's

done this before the hard part is

locating that direction they're not

going to give you the triangles usually

it's going to be in words so you have to

be able to draw the triangles and figure

out the directions on your own

so the first

first topic that we're going to look at

are the first way of doing directions is

based on the Cartesian system which is

what we use in math so those of you

doing math 20-1 you would have done this

already in the trigonometry unit

so what we do with the math directions

is we always start at the zero degrees

over on the right side then we go up to

90 degrees on the top 180

and 270.

so in physics if the question was you're

going to go 10 meters

at 70 degrees

then what we would do is that means

we're going to be drawing this thing

from the zero we measure 70 degrees

upwards

so that would be where our Vector would

be and then from there we could figure

out our

vertical and horizontal components if we

need to so that's the first one 10

meters at 70 degrees and for today we're

not going to worry about the 10 meters I

just want to worry about the

the where the location would be so the

next one is let's suppose I told you we

have something located at 220 degrees

so the 220

would be we'd go around to the 180 and

then we'd add on a little bit more and

that would get us

to 220. so the 220 degrees would look

something like that and then later on

when we do the trig calculations we're

not going to worry about the 220 we're

going to basically make our triangle

out of that and then we'd use 220 minus

80 we'd use 40 degrees

inside our triangle

okay so let's do a couple more so

wirewood

350 degrees be located well that's going

to be all the way around

just not quite to the 360 which would be

the zero location so that would be about

10 degrees like that or

350 all the way around okay so math

terms we call those the reference angle

so we're 350 degrees goes all the way

around our reference angle would be the

10 degrees that we're going to work with

and let's do one more let's suppose we

had an angle of

140 that's going to be

something like that

so our reference angle in this case

would be 40 degrees down there

okay so those of you that are doing this

in math this will seem familiar okay so

you'll find this this system pretty

basic and pretty easy you just have to

make sure you always go from the zero

and measure your way around

the next system we're going to look at

is probably I don't know if it's used

more than the other one but it's used

quite a bit in physics and this is the

navigation one and in this case this is

like a GPS system where we use

north east south and west instead of

the math system so in this case you're

going to get questions where they'll say

things like we have a vector going at

North

30 degrees east

okay so what you have to do is you have

to be able to figure okay what is that

where's that location so in this sense

the first letter is telling you we're

going towards the North First

and then we're going to be moving 30

degrees from there towards the east

so in terms of drawing our triangle it

would look something like that where 30

degrees would be down there so from the

north towards the east that would be

where our angle would be okay so let's

do a couple more of those so if I gave

you

South

65 degrees west

that means we're going to be moving from

the south towards the west and we're

going to go

65 degrees so it would look something

like that

all right if I gave you let's do one

more let's do

East

20 degrees south

so just similarly we'll be heading

towards the east first so from the East

towards the South we're gonna go

20 degrees

okay so that's sort of the navigation

system and they work pretty good if you

get a question that just says

um

Northeast only

then that means it's perfectly right

down the middle so that would be 45

degrees Northeast or 45 degrees east of

North so if you get two angles so let me

just write that one so if I go north 45

East

or east 45 North because they're both 45

degrees those are exactly the same thing

so there's no difference in either way

that's why we can just write it as

Northeast and it doesn't make any

difference

but if I gave you

let's do a different one if I gave you

North

30 degrees west

and I give you West 30 degrees north

those are not the same

okay those aren't the same because North

40 30 degrees west would be

from the north towards the west 30

degrees and west 30 degrees north would

be down here it would be 30 degrees

upwards from the West so those aren't

the same but what you could do if you

get one and you need to figure out the

other is instead of being North 30

degrees West and West 30 degrees north

all you'd have to do is change the other

one to the complementary angle so if

this angle up here is 30 that means the

one below it would be 60. so North 630

degrees West and West 60 degrees north

those are the same okay and it wouldn't

matter on a test if you're writing them

either one would be acceptable

there's one other thing we got to look

at when we're dealing with these I'll

just start a new page

so if we get a

Direction system like that in the

textbook and in other older materials

quite often they'll write the angles

instead of the way we just wrote them

they'll write them as 40 degrees

north of West

so in this case it says 40 degrees north

of West we're going from the West

towards the north north of West so in

this case that 40 degrees north and west

would be like that

and if you use the one that we just

looked at we said we're heading west

then going 40 degrees north so that

would be those two things are the same

so some people get mixed up with this

north of West South of East so if you

ever do that if I gave you a question

that was 35 degrees

south of East

if you want you can switch it to the

other around move the East to the front

see that East 35 South

and those two things would be the same

so East 35 South would be from the East

towards the South 35 degrees

or if you can understand 35 degrees

south of East

that's the same so either one both of

those systems are good you're going to

see questions where they're all mixed up

so you just have to be able to sort of

draw any kind of scenario

so if I gave you a different one that I

said

[Music]

60 degrees west

of North

or I told you to draw West 30 degrees

north or I try to tell you to draw 120

in the math Cartesian system you should

see that these are all the same thing so

60 degrees west of North

would be

60 degrees west of North would be like

that right from the north we're going 60

degrees towards the west or the other

one West 30 degrees north so from the

West we're going to head 30 degrees to

the north would be there and I messed up

on the last one it shouldn't be 120 it

should have been

150. so if we measure in the math system

150 degrees

would go all the way around and we'd get

to there so all three of those are the

same it doesn't matter which one you'd

be given you're going to see that

they're all exactly the same location so

any particular question could use any of

the three

locations