Physics 20: 2.2 Vector Components
Summary
TLDRThe 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
📚 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.
🚀 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.
🔍 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.
Mindmap
Keywords
💡Vectors
💡Right Angle Triangle
💡Resultant Vector
💡Pythagorean Theorem
💡Trigonometric Functions
💡Components
💡Uniform Motion
💡Significant Figures
💡Direction
💡Winds
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
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