In this video,
we will take a look at why the vector v
makes an angle theta with the positive x axis,
then a component form vector v,
has an x component equal to the magnitude of vector v
times cosine theta,
and a y component equal to the magnitude of vector v
times sin theta.
We'll explain this using what we already know
about right triangle trigonometry, shown here on the right,
and let's do this, we're working through an example.
We're asked to find the component form of a vector
with the magnitude of eight,
and forms a 60 degree angle with the positive x axis.
So to begin, I've already sketched this vector in blue
because the magnitude of the vector is eight.
We know the length of the vector is eight units.
And because the direction is 60 degrees,
or because the vector forms a 60 degree angle
with the positive x axis,
we know this angle here is 60 degrees.
For the next step, we can form a right triangle
using the horizontal and vertical components
of the given vector,
where this horizontal vector here
represents the horizontal component, which we can label x,
and this vertical vector here
represents the vertical component,
let's go and label this y.
Now if we find x and y using right triangle trigonometry,
we will also find the x and y components
of the given vector.
For a quick review, remember, when we have a right triangle,
cosine theta is equal to the ratio
of the adjacent side of the hypotenuse,
or if we are on the coordinate plane,
cosine theta is equal to X divided by r,
where r would be the radius of this circle here,
center to the origin,
which also notices the magnitude of the vector,
or the hypotenuse of the right triangle.
So if we have cosine theta equals x divided by r,
notice how we can solve for x
by multiplying both sides by r,
which gives us x equals r cosine theta.
And x equals r cosine theta,
is equivalent to x equals, the magnitude of vector v
times cosine theta.
Similarly, sin theta is equal to the ratio
of the opposite side of the hypotenuse,
on the coordinate plane,
sin theta is equal to y divided by r,
solving for a y here, we get y equals r sin theta,
and y equals r sin theta
is equivalent to y equals the magnitude of vector v
times sin theta.
To show this, going back to the right triangle
on the coordinate plane, notice how we can state
that cosine 60 degrees,
is equal to x divided by eight.
If we solve for x, we multiply both sides by eight,
which gives us x equals eight times cosine 60 degrees.
And we know that eight is the magnitude of vector v,
and therefore this equation here is equivalent
to x equals the magnitude of vector v, times cosine theta.
Similarly, using the right triangle,
notice how sin 60 degrees, is equal to y divided by eight.
Multiplying both sides by eight
gives us y equals eight sin 60 degrees.
Y equals eight sin 60 degrees,
is equivalent to y equals the magnitude of vector v
times sin theta.
Which is the reason why, the given vector, vector v,
has an x component of eight cosine 60 degrees,
and a y component of eight sin 60 degrees.
And now wishing to evaluate,
cosine 60 degrees and sin 60 degrees,
either using a reference triangle or the unit circle.
And let's review both, using the 30, 60, 90
reference triangle is shown here,
because we are in the first quadrant,
we know both legs will be positive,
and therefore, the cosine of 60 degrees
is equal to one divided by two or one half,
and sin 60 degrees is equal to,
square root three divided by two,
or the unit circle, the terminal side of 60 degrees
intersects the unit circle at this point here,
where the unit circle, x equals cosine theta
and y equals sin theta.
Verifying cosine 60 degrees is equal to one half,
and sin 60 degrees is equal to square root three
divided by two.
Which means the x component
is equal to eight times one half,
and the y one is equal to eight times
square root three, divided by two.
Simplifying, eight times one half is equal to four,
the x component is equal to four.
For the y component, that's right eight is a fraction
with the diameter of one, eight and two
share a common factor of two, there's one, two and two,
and four twos and eight,
the y component is equal to four square root three.
So in future videos, we will have to form right triangles
as we did here,
we can simply use the given formula as shown here.
I hope you found this helpful.