>> We want to write the vector with a magnitude of 8
in a direction of 3 degrees in component form.
This is the component form of a vector when the initial point is at the origin
and the terminal point is at the point X comma Y where X is the horizontal component
of the vector and Y is the vertical component of the vector.
So let's go ahead and sketch this vector on the coordinate plane
with the initial point at the origin.
The angle between the positive X axis and the vector would be 3 degrees.
And because the magnitude is 8, this vector would have a length of 8 units.
So this would be the horizontal component of the vector.
And this would be the vertical component of the vector.
So we'll call this X and we'll call this Y. And now because we have the right triangle,
we can solve for X and Y using trig equations.
For example, cosine 30 degrees is equal to the ratio of the adjacent side of the hypotenuse.
So we have cosine 30 degrees equals X divided by 8, which I'm going to write
as X divided by 8 equals cosine 3 degrees.
And then the sine of 3 degrees would be equal to the ratio
of the opposite side to the hypotenuse.
So the sine of 3 degrees is equal to Y divided by 8
or we could say Y divided by 8 equals sine 3 degrees.
In order to multiply both sides by 8, we have X equals 8 cosine 3 degrees.
Multiplying by 8 here, we have Y equals 8 times sine 3 degrees.
And these are actually the formulas that we use to find the component form of a vector.
X is going to be equal to the magnitude times cosine theta and Y is equal
to the magnitude times sine theta.
We're using our 30 60 90 right triangle here on the left.
Cosine 3 degrees is equal to square root 3 divided by 2 and sine 3 degrees is equal to 1/2.
So we have X equals 8 times square root 3 over 2.
This simplifies so X equals 4 square root 3.
And then sine 3 degrees is equal to 1/2, we have Y equals 8 times 1/2.
Of course this simplifies as well, so we have Y equals 4.
So the component form of our vector, which we'll call vector V,
is equal to 4 square root 3 comma 4.
Now that we've shown how to use right triangle trigonometry to find the component form
of a vector, given the magnitude and the direction, let's discuss the formulas
that we can use to save some time.
If we want the component form of vector U, and we have the magnitude and the direction,
X is going to be equal to the magnitude of U times cosine theta,
and Y is equal to the magnitude of U times sine theta.
And we actually used these 2 formulas in our first example,
but we derived them using our right triangle trigonometry.
We can also write the component form of vector in this form here, where we have the magnitude
of U times the component form of the vector, where we have cosine theta comma sine theta.
So for our second example, we'll actually use the formulas given here.
So because he magnitude is 14, and the direction is 135 degrees,
we would have X equals the magnitude 14 times cosine 135 degrees and Y is going to be equal
to the magnitude or 14 times sine 135 degrees.
If you take a look at our first example, notice how this was our second step,
once we formed the equations here using our right triangle.
Now to make sure we find the correct values for cosine 135 degrees and sine 135 degrees.
Let's go ahead and sketch this angle on the coordinate plane.
We'll be in the second quadrant.
Our reference triangle would be here.
Notice how the reference angle would be 45 degrees.
And because we have a 45 45 90 reference triangle, we can label it 2 legs 1
and the hypotenuse square root 2.
But because we're in the second quadrant, where X is negative, this leg will have
to be negative 1, Y is positive, so this leg stays positive 1.
Notice how this means the cosine function value is going to be negative.
So we'll have X equals 14, which I'll write as 14 over 1 times cosine 135 degrees,
using our reference triangle, is going to be negative 1 divided by square root 2.
We should recognize by now that if we rationalize a denominator,
this would just be negative square root 2 divided by 2.
So we have 14 times negative square root 2 divided by 2.
Of course this simplifies, so X is equal to negative 7 square root 2 and then Y is going
to be equal to 14 over 1 times the sine of 135 degrees is going to be equal
to positive 1 divided by square root 2, which is also equal to square root 2 divided by 2.
So this simplifies as well, to positive 7 square root 2.
So the component form of the given vector we'll call vector W,
is negative 7 square root 2 comma 7 square root 2.
I hope you found this 2 examples helpful.
I'll leave you with the formulas that we used in our second example. ------------------------------720bf50259f4--