Vector word problem: resultant velocity | Vectors | Precalculus | Khan Academy

- [Instructor] We're
told a boat is traveling

at a speed of 26 kilometers
per hour in a direction

that is a 300 degree rotation from East.

At a certain point it encounters a current

at a speed of 15 kilometers
per hour in a direction

that is a 25 degree rotation from East.

Answer two questions
about the boat's velocity

after it meets the current.

Alright, the first question is,

what is the boat's speed
after it meets the current?

And it says, round your
answer to the nearest 10th.

You can round intermediate
values to the nearest 100th.

And what is the direction
of the boat's velocity

after it meets the current?

And they say the same, well,
they actually here it say,

round your answer to the nearest integer

and you can round intermediate
values to the nearest 100th.

So like always pause this video

and see if you can work through this.

All right, now let's
work on this together.

So first let's visualize
each of these vectors.

We have this vector 26 kilometers
per hour in a direction

that is a 300 degree rotation from East.

And we have this vector
15 kilometers per hour

in a direction that is a 25
degree rotation from East.

And so let me draw some axes here.

So let's say that is my Y-axis.

And then let's say that
this over here is my X-axis.

And then that first vector
300 degree rotation from East,

East is in the positive X direction.

This would be 90 degrees,
180 degrees, 270 degrees.

I'm going counter-clockwise
cause that's the convention

for a positive angle.

And then we'd go a little bit past 270,

we would go right, right over there.

And the magnitude of this vector
is 26 kilometers per hour.

I'll just write a 26 right over there.

And then this other vector
which is the current

15 kilometers per hour in a direction

that is a 25 degree rotation from East.

So 25 degree rotation might
be something like this

and it's going to be shorter.

It's 15 kilometers per hour.

So, it's going to be
roughly about that long.

I'm obviously just approximating it

and I'll just write 15
there for its magnitude.

So we can visualize what the boat's speed

and direction it is after
it meets the current.

It's going to be the sum
of these two vectors.

And so if we wanted to
sum these two vectors

we could put the tail of one
at the head of the other.

And so let's shift this
blue vector down here.

So it's at the head of the red vector.

So it would be something like this.

And so our resulting speed
after it meets the current

would look something like this.

We've seen this in many
other videos so far

but we don't want to just
figure it out visually.

We want to actually figure
out its actual speed

which would be the
magnitude of this vector

and its actual direction.

So what is the angle?

And we could say it as a positive angle.

So what the rotation,
the positive rotation

from the positive X-axis or from due East.

So to do that, what I'm
going to do is represent each

of our original vectors in
terms of their components.

And so this red vector up here

and we've done this multiple
times explaining the intuition.

It's X component is
going to be its magnitude

26 times the cosine of this angle,

cosine of 300 degrees.

And it's Y component is
going to be 26 times the sine

of 300 degrees.

If that's unfamiliar to you,
I encourage you to review it

in other videos where we first introduced

the notion of components,
it comes straight out

of our unit circle
definition of trig functions.

And similarly, this vector right over here

it's X component is going
to be its magnitude times

the cosine of 25 degrees.

And it's Y component is
going to be 15 times the sine

of 25 degrees.

And now when we have
it expressed this way,

if we want to have the resulting vector,

let's call the resulting vector S

for maybe the resulting speed.

Its components are going to
be the sum of each of these.

So we can write it over here.

Vector S is going to be equal to

it's going to be the X
component of this red vector

of our original speed vector.

So, 26 cosine of 300 degrees

plus the X component of the current.

So, 15 times cosine of 25 degrees
and then the Y components.

Once again, I add the
corresponding Y components

26 sine of 300 degrees

plus 15 sine of 25 degrees.

And now we could use a
calculator to figure out

what these are, to say what
these approximately are.

So first the X component,
we're going to take

the cosine of 300 degrees, times 26,

plus I'll open a parenthesis here.

We're going to take the cosine
of 25 degrees, times 15,

close our parentheses.

And that is equal to 26.59 if
I round to the nearest 100th.

26.59.

And now let's do the Y component.

We have the sine of 300 degrees, times 26,

plus I'll open parentheses,

the sine of 25 degrees
times 15, close parentheses,

is equal to negative 16.18 to
round to the nearest 100th.

Negative 16.18.

And let's just make sure that
this makes intuitive sense.

So, 26.59.

So we're going to go
forward in this direction

26.59 on the X direction.

And then we go negative
16.18 in the Y direction.

So this does seem to match our intuition

when we tried to look at this visually.

So we now have the X and Y components

of the resulting vector

but that's not what they're asking for.

They're asking for the speed
which would be the magnitude

of this vector right over here.

And so I could write the
magnitude of that vector

which is going to be its speed.

We'll just use the
Pythagorean theorem here.

It is going to be the
square root of this squared

plus this squared, because once again

this forms a right triangle here.

And we review this in other videos,

it's going to be the square
root of 26.59 squared

plus negative 16.18 squared
which is approximately equal to,

they want us to round to the nearest 10th,

26.59 squared plus.

And it doesn't matter that
there's a negative here

cause I'm squaring it.

So I'll just write 16.18
squared is equal to that.

And then we want to take
the square root of that.

We get 31 point, if we round
to the nearest 10th, 31.1.

So it's approximately 31.1 and
we'll write the units here,

kilometers per hour is
the speed the boat's speed

after it meets the current.

And now the second question
is what is the direction

of the boat's velocity
after it meets the current?

Well, one way to think about it is

if we look at this angle right over here

which would tell us the direction

the tangent of that angle,
theta, let me write this down.

Tangent of that angle theta.

We know your tangent is your change in Y

over your change in X.

You can even view it as the slope

of this vector right over here.

We know what our changes in X or Y are.

Those are X and Y components.

So it's going to be our change in Y

which is negative 16.18, over 26.59,

our change in X.

And so to solve for theta, we could say

that theta will be equal
to the inverse tangent.

And we'll have to think
about this for a second

because this might not get us
the exact theta that we want

because the inverse tangent function

is going to give us something
between positive 90 degrees

and negative 90 degrees.

But the number we want, actually it looks

like it's going to be
between 270 and 360 degrees

because we're doing a,

we want to think about a positive rotation

instead of a negative one but
let's just try to evaluate it.

The inverse tan of this,

of negative 16.18, over 26.59.

16.18 negative

divided by 26.59 is equal to this.

And now I am going to take
the inverse tangent of that.

And that gets us negative 31
degrees, which makes sense.

This looks intuitive sense

that if you were to do
a clockwise rotation

which would be a negative
angle from the positive X-axis

it looks like what we
drew, but let's just go

with the convention of
everything else here.

And let's try to have a positive angle.

So what we can do is
add 360 degrees to that

to make a full rotation around.

And we essentially have
the equivalent angle.

So let's add 360 to that to
get that right over there.

So if we round to the nearest integer

we're looking at
approximately 329 degrees.

So theta is approximately 329 degrees.

So here, when I said theta is
equal to this I could write

theta is going to be equal
to this plus 360 degrees.

Now what's interesting is, I
was able to add 360 degrees

to get to the exact same place.

If we had a situation where
our angle was actually

this angle right over
here not the situation

that we actually dealt with,

but if it was in the second quadrant,

we would have gotten this theta.

And we would have had to
be able to realize that,

hey we're dealing with the second quadrant

that has the same slope.

So instead of adding 360 degrees

we would have added 180 degrees.

And we've also covered that
in other videos as well.