Physics 20: 2.3 2D Vectors
Summary
TLDRThis educational video script explains the process of solving vector addition in two dimensions with vectors at angles. The presenter guides viewers through the steps of determining vector components, combining them, and calculating the resultant vector. Using an example of a person walking 15 meters at 30° north of east and then 10 meters at 20° west of north, the script demonstrates how to find the horizontal and vertical components using trigonometry, add them to find the resultant displacement, and then use the Pythagorean theorem to determine the magnitude and direction of the final vector. The method is illustrated with clear examples, making it accessible for learners to understand and apply.
Takeaways
- 📐 Today, we are solving vector problems involving two vectors in two dimensions.
- 📊 First, break down the vectors into components to simplify the problem.
- 📝 Example problem: A person walks 15m at East 30° North and then 10m at North 20° West.
- 📏 Calculate the vertical and horizontal components of the first vector (15m at 30°).
- 📐 The vertical component is 7.5m and the horizontal component is 12.99m.
- 📏 Next, calculate the components of the second vector (10m at 20°).
- 📐 The vertical component is 9.4m and the horizontal component is 3.42m.
- 🔄 Combine the vertical components (7.5m + 9.4m = 16.9m) and the horizontal components (12.99m - 3.42m = 9.57m).
- 📐 Use the Pythagorean theorem to find the resultant vector length (19m) and angle (60°).
- ✅ Final displacement is 19m at East 60° North.
Q & A
What is the process for solving vector addition problems with two vectors in two dimensions?
-The process involves breaking down the problem into steps: first, determine the vector components for each vector, then combine these components, and finally calculate the resultant vector which includes its magnitude and direction.
How does the example in the script illustrate the vector addition process?
-The example demonstrates the process by showing a person walking 15 meters at 30° North of East, followed by 10 meters at 20° West of North. The script then guides through finding the horizontal and vertical components of each vector, combining them, and finally calculating the resultant vector's magnitude and direction.
What is the first step in solving for the vector components of a given vector?
-The first step is to sketch the situation and determine the horizontal (x) and vertical (y) components of the vector by using trigonometric functions such as sine for the vertical component and cosine for the horizontal component.
How are the horizontal and vertical components of the first vector in the example calculated?
-For the first vector, the vertical component is calculated using sin(30°) * 15 m, which equals 7.5 m. The horizontal component is calculated using cos(30°) * 15 m, which equals approximately 12.99 m.
What trigonometric functions are used to find the components of the second vector in the example?
-For the second vector, sine is used to find the horizontal component (sin(20°) * 10 m), and cosine is used to find the vertical component (cos(20°) * 10 m).
How are the components of the second vector combined with the first to find the resultant vector?
-The vertical components (9.4 m and 7.5 m) are added together, and the horizontal components (12.99 m to the right and 3.42 m to the left) are combined by subtracting the leftward component from the rightward component, resulting in a net horizontal component of 9.57 m to the right.
What mathematical theorem is used to find the magnitude of the resultant vector?
-The Pythagorean theorem is used to find the magnitude of the resultant vector by adding the squares of the combined vertical and horizontal components and then taking the square root of the sum.
How is the direction of the resultant vector determined?
-The direction of the resultant vector is determined by calculating the angle it makes with the horizontal axis, which can be found using trigonometric functions and the components of the resultant vector.
What is the final answer for the magnitude and direction of the resultant vector in the example?
-The final answer for the magnitude is approximately 19.4 meters, and the direction is East 60° North.
Why is it important to break down the original vectors into their components before combining them?
-Breaking down the original vectors into their components simplifies the process of vector addition, especially when the vectors are not aligned with the coordinate axes. It allows for easier calculation of the resultant vector's magnitude and direction.
Can the process demonstrated in the script be applied to vectors with different magnitudes or angles?
-Yes, the process can be applied to vectors with different magnitudes or angles. The key is to calculate the horizontal and vertical components for each vector and then combine and recalculate the resultant vector accordingly.
Outlines
📚 Introduction to Solving Vector Addition in Two Dimensions
This paragraph introduces the concept of solving vector addition problems in two dimensions where vectors are at specific angles, making the calculations more complex. The speaker outlines the process of breaking down the problem into manageable steps: identifying vector components, combining them, and determining the resultant vector. An example is presented where a person walks 15 meters east at 30° north and then 10 meters at 20° west of north. The goal is to find the final displacement and direction. The speaker emphasizes the importance of sketching the situation and calculating the vertical and horizontal components of each vector using trigonometric functions like sine and cosine. The first vector's components are calculated as 12.99 meters east and 7.5 meters north.
🔍 Calculating Components and Resultant Vector for Two Vectors
In this paragraph, the process continues with the calculation of the second vector's components. The horizontal component is found using the sine of the angle (20°), resulting in 3.42 meters to the west, and the vertical component using the cosine of the angle (20°), resulting in 9.4 meters upward. The speaker then combines the vertical components of both vectors, summing 9.4 meters and 7.5 meters to get a total upward displacement of 16.9 meters. For the horizontal components, the eastward displacement (12.99 meters) is adjusted by subtracting the westward displacement (3.42 meters), yielding a net eastward displacement of 9.57 meters. The final step involves using the Pythagorean theorem to find the magnitude of the resultant vector, which is approximately 19.43 meters, and calculating the direction, which is found to be 60° north of east. The paragraph concludes with the final answer presented in vector notation, rounded to two decimal places.
Mindmap
Keywords
💡Vector Components
💡Resultant Vector
💡Trigonometry
💡Pythagorean Theorem
💡Displacement
💡Direction
💡Angles
💡Horizontal Component
💡Vertical Component
💡Sine and Cosine Functions
Highlights
Introduction to solving vector component questions with two vectors in two dimensions.
First step: figure out the vector components and then combine them.
Example problem: adding vectors 15 meters at East 30° North and 10 meters at North 20° West.
Sketching the situation to visualize the problem.
Finding the resultant displacement and direction.
Breaking down the vectors into vertical and horizontal components.
Calculating the components using trigonometric functions: sine and cosine.
Summing up vertical components: 7.5 (first vector) + 9.4 (second vector) = 16.9.
Calculating the horizontal components: 12.99 (first vector) - 3.42 (second vector) = 9.57.
Using Pythagorean theorem to find the resultant vector's magnitude: 19.43.
Finding the angle of the resultant vector: 60°.
Stating the final vector in terms of magnitude and direction: 19 meters at East 60° North.
Second example problem: 25 meters/second at an angle of 40° from Northwest.
Second vector: 30 meters/second straight South.
Repeating the process of breaking into components and solving.
Transcripts
okay today we're going to look at
solving Vector com questions that have
uh two vectors in two Dimensions so both
of them are going at Angles so that
makes things a little more complicated
but what we have to do is we just have
to break it down into a couple of steps
where we do exactly what we did before
so we're going to first of all figure
out what the vector components are and
then we're going to combine those and
then figure out what the resultant would
be so the best way to show this is just
to start with an example so let's do
that so let's our question is so a
person walks 15
M
at so let's suppose we want to add 15
M at uh
East 30°
north and then let's say they're going
to go 10 m after
that
at
North
20° West okay so the question is what is
the resultant what is our final
displacement and Direction so for all
these questions you want to find the
length of the vector plus the
direction okay so our first the first
thing is let's sketch out this this
situation so let's suppose we started at
the bottom here so we're going to go at
30
Dees okay so we know it's going to be
30° from east to North nor and this was
15 and then afterwards the second Vector
we're going to go from head to tail so
we're going to start the second one and
it's going to go something like
that right where this is
20° and that one's going to go 10 so the
question then is what is the resultant
so our resultant would be from
start to finish okay so for us to try to
do this with trigonometry it's a little
bit difficult because we don't know all
the angles we don't know whether it's a
right angle triangle or not so the best
bet is to first step is to figure out
the components so if we look at just
that first Vector the
15 the first thing you want to do is
figure out the vertical and horizontal
components of of that vector by itself
okay so we know it's 30° we know it's 15
so we can easily solve for each of those
so our vertical side because the Y is
the opposite so you'd have S of 30
equals opposite over
15 and we can do that on our calculator
so just go sin
30 time 15 so that'll give us
7.5 okay so we know that that side is
7.5 and then we want to do the same
thing for the bottom but the bottom this
time is adjacent so we'd have cosine of
30 = x over
15 so do that one on your calculator
go cos 30 *
15 and that gives us
12.99 we'll leave the decimals intact
for now we'll round off at the end okay
so 12.99 so we've got our two sides of
that initial triangle let me write those
in there so we have 12.99 to the East
and then our vertical component was 7.5
to the
north okay so now what we want to do is
basically repeat the exact same thing
using the second triangle so our second
triangle we're going to have a vertical
component that goes up first and then
our horizontal goes to the left okay so
we want to basically do the exact same
thing okay so we want to find the
horizontal and vertical components in
this case the horizontal is our opposite
side
so we'd
have sin
20 okay so we got the opposite side so
sin 20 would equal x over 10 and then
the same thing for the vertical
component but it'll be cosine because
it's our adjacent so cos 20 = y/ 10 so
that's our calculations we need for the
second triangle so let's solve for each
of those So Co or that's to the S 21st
so s 20 *
10 gives us
3.42 and then do cos 20 *
10 gives us
9.4 if we round it
off okay so let's write those on our
original triangle again so we have 3
42 on the
top and 9.4 on the
side and that's it so we've got our
triangles calculated so we figured out
all the components for the two vectors
so now when you look at it you can see
basically what we do have is we have two
vectors that go up right we have our two
vertical vectors so if we can combine
those two into one so let's just make
one tall Vector that goes up and it'll
be
9.40 plus 7.5 cuz they're both going
upward so we can add them together so
that would be the same thing as
[Music]
16.9 and then if you look at the
horizontal vectors we got 12.99 to the
right but then we have
3.42 going to the left so because those
are in opposite directions we need to
subtract them so
12.99 - 3.42 2 gives us an answer of
9.57 to the right that's the bigger
Direction okay so what we want to do
then let's just put the two vectors head
to tail it doesn't matter whether I drew
it like this or I could draw the 9.57
starting up here and go this way it
really doesn't make any difference I
just picked the bottom because that's
going to look similar to our original
right so now the question is what is the
resultant so we want to find out what
that side and angle is okay so when you
look at these two triangles you can see
that our original resultant right there
should be the exact same as the green
resultant that we're going to solve for
now okay so all we have to do for the
green one is just like we did with the
other questions find that length and
that angle and we're done so the length
is pretty easy we just use Pythagorean
theorem so 16.9 SAR + 9.57
squar and square root that answer
that gives us
[Music]
19.429661
[Music]
that and we get an angle of
60° so that's it we've done the question
so the final step then would be just to
State your answer so we'll just State
our answer in final Vector notation and
let's round it off to two digits
everything should be in two digits so
our final answer was 19 M right the 19.4
we'll round it to two and our Direction
was 60° so we went East
60°
North and that's it okay so you can see
with this question we sort of did two
steps First Step was break the original
vectors into its two components then
combine those into one new triangle and
then solve that new triangle for the
hypotenuse and the
angle let's try one more and that'll be
it for today so let's suppose in this
case I've got a vector that
goes 25 m/ second at an angle
[Music]
of let's suppose that angle is 40° from
Northeast or north west and then let's
suppose our second Vector then
goes
down
at um let's say it's 30
m/s and that one is straight South okay
so this one is perfectly so the other
one was at an angle of
40° so we want to do the exact same
thing again so let's break it into our
horizontal and vertical
components so that first first Vector
we'll just do those two
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