mr.p: Good morning.
Today we are going to learn about
component vectors by breaking a
displacement vector into its vector components.
Here we go!
Bo: Hey guys.
Billy: Hey, Bo.
Bobby: Hi, Bo.
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mr.p: We're going to start by letting
our slow velocity racer go through
a displacement of 90.0 centimeters
at an angle of 32 degrees north of east.
Now, this displacement, which we will label "d"
can be broken into its component vectors
displacement in the x direction,
or d sub x and displacement in the y direction,
or d sub y.
Bo, how can we find the components of vector d?
How can we find d in the y direction
or d in the x direction?
In other words, how can we find either of these
two sides of the triangle?
Bo: Well, this is a right triangle so...
mr.p: Sure Bo, you can use a SOH.
Bo: So sine theta--
Billy: Yup, theta.
Bo: So sine theta equals opposite over hypotenuse
and opposite the angle is d in the y direction.
The hypotenuse of the triangle is d,
we wanna solve for d in the y direction
so multiply both sides by d,
so you get d in the y direction
equals d times the sine of theta.
At this point we can substitute in numbers
so we get 90 times the sine of 32 degrees or--
Bobby: 49.628 centimeters.
Bo: Actually no, it's 47.693 centimeters.
Bobby: Uh, my calculator says 49.628
and my calculator is never wrong.
And you don't even have a calculator.
Bo: Well actually your calculator
is in radian mode.
In order to do this you need your calculator
in degree mode because our angles are in degrees,
not radians.
Bobby: Yeah, you're right.
Thanks.
Bo: You are welcome.
mr.p: Correct, we can use this fact that
sine theta equals opposite of our hypotenuse
to solve for our displacement in the y direction.
And because our angle of 32 degrees
only had two significant digits,
we should have two in our answer
for 48 centimeters for the displacement
in the y direction.
And Billy, how can we figure out the displacement
in the x direction?
Billy: Now we can use cosine theta equals
adjacent over a hypotenuse,
or displacement in the x direction over d.
Multiplying by d,
we get the displacement in the x direction
equals d times a cosine of the angle,
with numbers that works out to be
90 times the cosine of 32 which is--
Bobby: 76.324 or with two sig figs,
76 centimeters.
mr.p: Correct, you can use the fact that
cosine of theta equals adjacent over
a hypotenuse to solve for
the displacement in the x direction
which is 76 centimeters.
So dx and dy are the components of vector d
and what we did was to break vector d into
those components.
Some people prefer to say resolve the vector
into those components,
but I prefer to say break the vectors
into components because that hard K, K,
is so much more fun to say. (laughs)
Bo: Break.
Billy: Weird.
mr.p: It is important to understand
what it means that d sub x and d sub y
are the components of vector d.
This means that when you add vector d
in the x direction to vector d in the y direction,
you get the vector d.
We can show this visually using our
slow velocity racer.
First we can have her travel the component
that is the displacement in the x direction,
or 76 centimeters in the positive x direction.
And then we can turn Slow Velocity Racer
90 degrees and have her travel the component
that is the displacement in the y direction,
or 48 centimeters in the positive y direction.
And notice that she ends in the exact same spot
as the original displacement vector d.
Again, this is because the components
of vector d, d sub x and d sub y,
add up to the original vector d.
Which means that vector d is the resultant vector.
Bo: Hold up Mr. P.
Won't Slow Velocity Racer be facing
in a different direction when she gets
to the final point?
I don't see how it's the same thing then.
mr.p: Good question, Bo.
We actually have to go back
to the definition of displacement
in order to answer your question.
So Billy, what is the definition of displacement?
Billy: Displacement is the straight line distance
between the initial and the final points.
Bobby: It's also the final position
minus the initial position.
Bo: Yeah, I remember now.
Displacement only depends on the initial
and the final points.
The path is irrelevant.
So the direction the object is facing doesn't matter.
I got it.
mr.p: Good.
Now, we can show that d sub x and d sub y
add up to d mathematically
by doing this problem in reverse.
What we're gonna do is we're gonna start out by
figuring out the magnitude of vector d
by using the Pythagorean Theorem.
A squared plus b squared equals c squared.
In other words, the displacement
in the x direction squared
plus the displacement in the y direction squared
equals the displacement squared.
Taking the square root of both sides,
we get the magnitude of the displacement
equals the square root of the displacement
in the x direction squared
plus the displacement in the y direction squared.
Substituting in numbers,
we get 76.324 squared plus 47.693 squared.
The square root of that whole thing
works out to be 89.894 centimeters,
the magnitude of the displacement.
And now we can figure out the angle
using tangent.
Tangent of theta equals opposite over adjacent
opposite our angle theta
is the displacement in the y direction
adjacent is the displacement in the x direction.
Taking the inverse tangent of both sides,
we get that the theta is equal to the
inverse tangent of the displacement in the y direction
divided by the displacement in the x direction.
Substituting in numbers,
theta is equal to the inverse tangent of
47.693 divided by 76.324.
And theta then works out to be 31.892 degrees.
With two sig figs because our original angle
only had two significant figures,
we get our displacement of 90 centimeters
or 9.0 times ten to the first centimeters
at an angle of 32 degrees north of east,
which is the same as our original displacement vector
with two sig figs because again,
the components of vector d add up to
the original vector d.
That's what it means to be the components
of vector d.
Bobby: OK that makes sense.
But I still have a question about
the displacement in the x and y directions.
I thought displacement was a vector
and had both magnitude and direction.
So d sub x is 76 cetimeters.
Wouldn't that just be the magnitude
without the direction?
mr.p: That's a great question, Bobby.
Now this displacement in the x direction
of 76 centimeters actually is a vector
and has both magnitude and direction.
Here is how:
This x means it's in the x direction
and the fact that number is positive
means it's in the positive x direction.
So the displacement in the x direction
of 76 centimeters literally means that this is
a displacement of 76 centimeters
in the positive x direction.
Bo: So d sub y equals 48 centimeters means
it's a displacement in the positive y direction
of 48 centimeters.
Yeah, I can see how that shows
both magnitude and direction.
mr.p: Great.
Just so you know, this concept of
breaking or resolving a vector into its components
is a concept you're gonna use for the rest
of your physics careers.
No matter how long they last.
Any more questions?
Bo: Nope, I'm good.
Billy: Not right now.
mr.p: Then on to the review.
We used sine and cosine to figure out
the pieces of vector d,
or the components of vector d.
Literally the displacement in the x direction
plus the displacement in the y direction
equals the original displacement.
Because that's what it means to be
a component of a vector.
You are a piece of the original vector.
Thank you very much for learning with me today.
I enjoyed learning with you.