Physics 20: 2.2 Vector Components

Summary

The video tutorial covers the basics of vectors, focusing on how to draw and solve them using right-angle triangle math. It provides step-by-step examples to demonstrate calculating the resultant vector and angle using Pythagorean theorem and trigonometric functions. The instructor emphasizes careful diagram drawing and explains the importance of maintaining significant figures. The lesson also covers reverse calculations to find vector components and includes various methods to express the final answers in vector form.

Takeaways

• 📏 Vectors can be represented and solved using basic right-angle triangle math.
• 🚗 Example 1: A car traveling east at 15 m/s and south at 11 m/s requires finding the resultant vector.
• 🔍 Resultant vectors are found by drawing a right-angle triangle and calculating the hypotenuse and angles.
• 📐 Use Pythagorean theorem to find the resultant magnitude: \(√(15^2 + 11^2) = 18.6\) m/s.
• 🔢 Calculate the angle using trigonometric functions: \(tan^{-1}(11/15) = 36.3°\).
• 🧭 Direction can be expressed in terms of cardinal directions or degrees: 36.3° south of east or 324°.
• ✈️ Example 2: An airplane heading north at 32 m/s with an east wind of 12 m/s requires similar steps to find the resultant vector.
• 📊 Apply Pythagorean theorem and trigonometric functions for second example: 34 m/s at 21° east of north.
• ↩️ Reverse process: Decomposing a vector into its horizontal and vertical components using sine and cosine functions.
• ⚖️ Example 3: A vector of 16 m at 27° east of north is decomposed into 7.26 m east and 14.3 m north.
• 🚀 Consistency in angles and components is key for accurate vector analysis and representation.

Q & A

What is the main topic discussed in the video script?

The main topic discussed in the video script is how to properly draw and calculate vectors, specifically focusing on velocity vectors and their components.

Why is it important to be careful with diagrams when dealing with vectors?

It is important to be careful with diagrams because they help visualize the direction and magnitude of vectors, which are crucial for accurately performing mathematical operations and solving vector problems.

What is the first example given in the script involving vectors?

The first example given in the script involves a car traveling with a uniform motion where the East component of the motion is 15 meters per second and the South component is 11 meters per second.

How is the resultant velocity vector calculated in the car example?

The resultant velocity vector is calculated using the Pythagorean theorem, by adding the squares of the East and South components, and then taking the square root of the sum to find the magnitude of the resultant vector.

What mathematical operation is used to find the magnitude of the resultant vector?

The Pythagorean theorem is used to find the magnitude of the resultant vector by calculating the square root of the sum of the squares of its components.

How many significant digits should be kept in the final answer according to the car example in the script?

The final answer should be kept to three significant digits, as per the given values in the car example (15.0 and 11.0).

What trigonometric function is used to find the angle when you have the opposite and adjacent sides of a right triangle?

The tangent function (tan) is used to find the angle when you have the opposite and adjacent sides of a right triangle.

How is the angle of the resultant vector found in the car example?

The angle of the resultant vector is found by using the tangent function with the South component (opposite side) over the East component (adjacent side), and then using the inverse tangent function (second ten on a calculator) to find the angle in degrees.

What is the process for finding the components of a vector when given its magnitude and direction?

The process involves using trigonometric functions sine and cosine to find the horizontal (east/west) and vertical (north/south) components of the vector, respectively.

How are the components of a vector expressed in terms of direction?

The components of a vector are expressed in terms of direction by stating the magnitude followed by the direction, such as 'North' or 'East', or by giving the angle from the horizontal or vertical axis.

What is the significance of the wind direction mentioned in the airplane example?

The wind direction is significant because it affects the airplane's resultant velocity. A wind blowing from the West (Westerly wind) means it is blowing towards the East, which is an important factor when calculating the airplane's actual velocity.

How is the resultant velocity of the airplane calculated in the script?

The resultant velocity of the airplane is calculated using the Pythagorean theorem with the northward velocity and the eastward wind velocity as components, and then finding the angle using the tangent function.

What are the two components of a vector and why are they important?

The two components of a vector are its horizontal (east/west) and vertical (north/south) parts. They are important because they allow for the analysis of a vector's effect in different directions, which is useful in various applications such as physics and engineering.

How does the script suggest finding the horizontal component of a vector?

The script suggests finding the horizontal component of a vector by using the cosine of the given angle and the magnitude of the vector, calculated as the magnitude times the cosine of the angle.

How does the script suggest finding the vertical component of a vector?

The script suggests finding the vertical component of a vector by using the sine of the given angle and the magnitude of the vector, calculated as the magnitude times the sine of the angle.

Outlines

00:00

📚 Introduction to Vectors and Basic Math

This paragraph introduces the concept of vectors, focusing on how to properly draw them and perform the necessary math to solve problems related to them. It emphasizes the importance of careful diagramming and uses the example of a car traveling eastward with a velocity vector. The math involved is based on right-angle triangles, specifically using the Pythagorean theorem to find the resultant vector magnitude and trigonometric functions to find the direction. The example provided walks through calculating the magnitude and direction of the resultant velocity vector of a car moving east and south, resulting in an 18.6 m/s velocity at an angle of 36.3 degrees south of east.

05:02

🚀 Calculating Resultant Velocities with Vectors

The second paragraph continues the discussion on vectors, specifically addressing how to calculate the resultant velocity when given component velocities. It uses the example of an airplane flying north with an eastward wind, illustrating the process of combining these vectors using the Pythagorean theorem to find the magnitude and trigonometric functions to find the direction. The resultant velocity is calculated to be 34 m/s at an angle of 21 degrees east of north. The paragraph also addresses the importance of understanding directional terms when dealing with wind vectors and provides alternative ways to express the final vector direction.

10:02

🔍 Finding Vector Components Using Trigonometry

The final paragraph shifts the focus to determining the components of a given vector, which are its horizontal and vertical parts. It uses the example of a vector described by its magnitude and angle relative to north, and explains how to work backward to find the north and east components using sine and cosine functions. The example calculates a horizontal component of 7.26 meters east and a vertical component of 14.3 meters north. The paragraph also touches on the importance of drawing accurate diagrams and understanding trigonometric relationships to solve for vector components.

Keywords

💡Vectors

Vectors are mathematical objects that have both magnitude (length) and direction. In the context of the video, vectors are used to represent physical quantities such as velocity, with components in different directions. For example, the video discusses velocity vectors with east and south components to describe the motion of a car.

💡Right Angle Triangle

A right angle triangle is a geometric figure with one angle measuring 90 degrees. In the video, right angle triangles are formed when dealing with vectors at 90-degree angles to each other, such as east and south components of velocity. The Pythagorean theorem is applied to these triangles to find the resultant vector magnitude.

💡Resultant Vector

The resultant vector is the single vector that represents the combined effect of two or more vectors. In the script, the resultant vector is calculated to determine the overall velocity of a car moving east and south, or an airplane flying north with an eastward wind.

💡Pythagorean Theorem

The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The video uses this theorem to calculate the magnitude of the resultant vector from its components.

💡Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are used to relate the angles of a triangle to the ratios of its sides. In the video, these functions are applied to find the components of a vector when given its magnitude and direction, as seen when determining the north and east components of a vector pointing 27 degrees east of north.

💡Components

Components refer to the individual parts of a vector along the horizontal and vertical axes, or along any other pair of perpendicular directions. The script explains how to find these components using trigonometry when given a vector's magnitude and the angle it makes with a reference direction.

💡Uniform Motion

Uniform motion is a type of motion where an object moves at a constant speed and in a straight line. In the video, the concept is used to describe the eastward motion of a car, which is a component of the car's overall velocity vector.

💡Significant Figures

Significant figures are the digits in a number that carry meaningful information about its precision. The video emphasizes the importance of keeping the number of significant figures consistent when performing calculations, such as rounding the resultant vector magnitude to three significant figures.

💡Direction

Direction is a crucial aspect of vectors, indicating the line along which the vector acts. The video discusses how to express direction in terms of angles, such as 36.3 degrees south of east or 69 degrees west of north, when describing the resultant velocity vector.

💡Winds

In the context of the video, winds are considered as vectors that affect the motion of objects like airplanes. A westerly wind is described as blowing from the west, which is an eastward component that needs to be taken into account when calculating the resultant velocity of the airplane.

Highlights

Introduction to vectors and their proper drawing and mathematical solutions.

Vectors are broken down into basic right angle triangle problems.

Importance of careful diagram drawing in vector calculations.

Example of calculating the resultant velocity vector from East and South components.

Use of Pythagorean theorem to find the magnitude of the resultant vector.

Explanation of how to find the angle of the resultant vector using trigonometric functions.

Finalizing the vector magnitude and angle while maintaining significant digits.

Expressing the final vector in both words and mathematical terms.

Handling wind as a vector affecting the resultant velocity of an airplane.

Calculating the resultant velocity of an airplane with a headwind.

Understanding the difference between wind direction and its effect on motion.

Finding the components of a vector given its magnitude and angle.

Using sine and cosine to calculate the horizontal and vertical components of a vector.

Diagramming the vector components for clarity and accuracy.

Example of calculating the components of a vector thrown at an angle.

Flexibility in choosing the order of drawing vector components.

Conclusion and预告 of future lessons on multiple vector components.

Transcripts

okay we're gonna look at vectors today

and we're just going to figure out how

to draw these properly and then do the

math to solve them you're going to see

that the math that we do is just basic

right angle triangle so it's pretty

straightforward

and uh

so the the math is going to be simple

you just got to be really careful with

your diagrams so the first example I'm

just going to do a few questions off the

worksheets the one that has the velocity

vectors the first one that's down below

it says a car is traveling in a straight

line with uniform motion the East

component of the motion is 15. so what

you want to do is make sure you draw

that properly so draw your vector at

15 meters per second

and then it says the South component is

11. so when you're drawing vectors you

always want to draw so draw your first

one put the arrowhead so you know which

direction it's going so this one's going

east then you want to start your second

one from where that one ended off

and then go south so in this case we'd

have 15 East and then 11 South so the

question is what is the velocity of the

vector or what is the resultant so when

we do resultant that's always going to

be from where we started to where we

ended so the resultant in this case

is the blue line

okay so that'll be what you actually

are moving so instead of saying I went

East and then South we could have just

went at an angle in the straight line

that the resultant shows

so when you're doing these you can see

it's just a basic 90 degree angle

triangle but because it's vectors we

actually have to find the length of the

sides we have to find that side there

and we actually have to find the missing

angle so you have to do both so the

first part of it let's do the side so

you can see because it's a right angle

triangle you can just do Pythagorean

theorem so X

will equal the square root right it's a

squared plus b squared so 15 squared

plus b squared would be 11 squared and

then just square root your answer

so when you go 15 squared plus 11

squared

do that on your calculator and you get

346 square root that answer and we get

18.6 go back to the question that the

original question was 15.0 and 11.0 so

we had three Sig digs so we should keep

our final answer in three Sig digs as

well so we have 18.6 meters per second

for our Vector size but like I said

before we have to find that angle so

when if you look at the angle

you can see we've got our opposite side

is 11 and our adjacent side is 15 and we

just calculated the hypotenuse so

technically we could use

any of the sine cos or tan but just like

before it's better to not use your

answer you calculated just in case you

made a mistake so in this case if we

were to just use the

the one that we know we pick 10 so we'd

have 10 of the angle

is opposite over hypotenuse 11 over 15

and remember when you're looking for an

angle

the way to solve it is you have to do

second ten so on your calculator go

second tan

11 divided by 15.

and that's it so if you go shift

shift 10 11 divided by 15. you should

get an answer of 36

.3 if we round it off to three Sig digs

so

36.3 degrees

okay so that's good we've got everything

figured out but the only there's one

more step that you have to do and that

is actually write your answer in final

Vector form so we'd have 18.6

meters per second

[Music]

and then we say what direction it's at

so our answer was 36.3 so you just got

to add in the north south east west or

you could use the math version that we

did the other day so in this case if we

would let's do the north south east west

first we went East so we started off

going east

then we went down at an angle of 36.3

degrees

towards the South

okay so that's one of the answers that'd

be acceptable or if you wanted to you

could do the math way so it'd be 18.6

meters per second

and for an angle remember it's from the

zero line all the way around so you'd

have to go zero all the way around to

that angle so you'd actually have 360

minus 36. so that would give us

three

three twenty three point seven but we

rounded to three Sig digs so we'd have

324 degrees

okay so either of those two answers

would be acceptable for this question

let's try one more let's do uh let's do

look question number three that's on

there so it says an airplane is headed

due north at a speed of 32.

Okay so we've got one that's going

straight North 32.

and it says a wind is arising from the

West which means it's blowing East so be

careful with these kind of questions

when you see them whenever they say a

wind they say a Westerly wind that means

it's from the west or if they said you

have a East Wind that means it's going

from the East blowing towards the West

so just be a little bit careful when you

do those directions so this one we have

32 to the North 12th to the east so the

question is what is the resultant

velocity

so we want to go from start to finish

just like that so we want that side

and that angle so we just do the exact

same thing we just did

so to get the side we'd go Pythagorean

theorem so 32

squared plus 12 squared

square root that answer

and that should give you 34 it works out

too

okay and then for the angle

go second ten

of 12 divided by 32.

and that should give you

we're just doing two Sig digs in this

case so you should get 21 degrees

okay

so those are our two answers so now just

like we did before you want to make sure

you write them properly so you have 34

original question was in meters per

second so we'd have 34 meters per second

and our angle would be

North

21 degrees

East

okay or remember we could use it the

other way we could write it as 21

degrees east of North

or if we wanted to use the math version

it would be from the zero lineup so it'd

actually be 90 degrees minus the 21

which would be

69 degrees

okay so any of those three directions

would be acceptable

okay and you could actually do other

ones you could say East 69 North that

would be fine too so a few different

ways of expressing answers it really

doesn't matter just pick the one that

you're comfortable with and go from

there

the next thing we'll look at so we'll

continue with this is what happens if

you get questions that are in the

reverse order so this is the second

worksheet the one that says components

of vectors so people tend to have a

little bit more trouble with going

backwards so what you want to do with

these is make sure you draw your

original vectors

perfect so to start with so we're going

to look at one B so it says 16 meters

16.0 meters at 27 degrees

east of North

okay so there's the question and what

we're looking for what are its

components

so when they say components what they

mean is what are the horizontal and

vertical components so the parts that

are going north or south and the parts

that are going east or west so we want

to basically take the answer now and

work backwards so with these ones just

make sure you draw them correctly so

let's sort of do a little grid so we

kind of starting in the right direction

so if we're going east of North that

means we should be going

something like that so that 27 degrees

would be down there

the length of our line is 16.0

so our components would be

a North component and then we should

have our East component like that so

east of North means to the north first

and then to the east okay so there's our

diagram so now all we have to do is

figure out what each of those sides are

so you can see here we're just using

regular trig again except this time now

we have opposite and adjacent

so let's call our adjacent side the

vertical Side Y because that's typically

what we use and then for our horizontal

let's call it X

okay so we just want to find each of

these so you can see from the angle we

have opposite and hypotenuse we can just

use regular sine so sine 27

is opposite over hypotenuse X over 16.

so to solve that all we have to do is go

16 times sine 27

so do that on your calculator

and you get

if we round it to three Sig digs because

that's what the question was there was

actually a 27.0

so everything had three Sig digs so we

should get

7.26 meters

and that was our horizontal component

that's the one that was going east so

let's add

East on there

okay then we do the same thing for our

vertical component so we have the

adjacent side this time so you're going

to use cosine so cos 27 will equal

y over 16.

so we'll cross multiply those ones so

you'd have

16 times cos 27

. so when you do that on your calculator

we get 14.26

so we'll round it to three Sig dig so

just 14.3 and that one is

the one that's going vertically and

we're heading north so we'd write it as

North and that's it so when it asks for

its two components that's all you have

to do is do the sine and cosine for each

and make sure you got the east or west

figured out and the North or South

figured out

so let's do one more let's look at

it's we'll look at number three so the

ball is thrown into the air at an angle

of 40 degrees to the horizontal

so 40 degrees would be

something like that so there's our 40

degrees

so the question is

it's or hold on it's saying it's going

at 25.0

meters per second and or was 40.0

degrees so we have three Sig digs so the

question is what are its components so

we want to find the vertical or and the

horizontal so the question is should we

draw it East First

and then North

or should we go north and then East and

it really doesn't matter it'll work the

same either way you just got to pick one

and make sure you have the right angle

so in this case the way I drew the 40

degrees it's better to draw the triangle

like that if we wanted to do let me just

show you the other version if we drew it

like that to be the 25 degrees and we

wanted to draw

the vertical and horizontal to look

something like that then the only thing

that would be different is we'd have to

figure out what that other angle is so

if it's 40 degrees up that means we'd

have 50 degrees there and there's 25

it'd still be there and we'd get the

exact same

solution so it doesn't matter which one

you pick just make sure you have the

angle properly calculated so for this

one let's do our horizontal component

first so it's our adjacent side so we'd

have cos 40

equals x over 25.

so when you do that one you should get

25 times cos 40.

gives us 19.2 if we round it off to

three Sig digs so 19.2 East

and then do the same thing for the Y

component

the vertical components we'd have sine

oops sine 40

equals y over 25

cross multiply those

and for y we should get 16.1

and I forgot to write the unit so it

should be 16.1 meters per second

and this one is going

North

okay so on the previous one I should

have had meters per second

as well 19.2 meters per second heading

east and those would be your two answers

that would be all that you have to do

for the question

okay so that's it for the components

Let's uh

we'll stop there and then next week

we'll continue when the we're going to

be doing components of more than one

vector at a time