Find the component form
of the vector v if the magnitude
is equal to a nine square root 85,
and when drawn in standard position vector v lies
in quadrant I and makes an angle measuring
arc-tangent 9/2 with the positive x-axis.
So we know the vector's in the first quadrant.
Let's say this is vector v.
And because the magnitude is nine square root 85,
we know the length of the vector is nine square root 85.
And now to form a right triangle
by sketching the horizontal and vertical components
of the given vector.
So this would be the x component, or horizontal component.
And this would be the vertical, or y, component.
And the angle theta is the angle measured
from the positive x-axis
to the vector rotating counter-clockwise, this angle here.
Which we know measures arc-tangent 9/2.
So using this reference triangle,
we might be thinking we can label the opposite side nine,
and the adjacent side two.
But that's not true, because it could be any multiple
of the ratio of nine to two.
So let's label the opposite side nine n,
and the adjacent side two n.
If we try to use two and nine,
two squared plus nine squared does not
equal nine square root 85, and that would be incorrect.
And now if we can find two n and nine n,
we can determine the x and y components
of the given vector.
And we can do this by applying the Pythagorean theorem
again, because we do have a right triangle.
Where the Pythagorean theorem
is c squared equals a squared plus b squared,
c squared is the square of the hypotenuse,
which is the square of nine square root 85.
And a squared plus b squared
is the sum of the squares of the legs.
It does not matter which one is a, which one is b.
So we have the square of two n
plus the square of nine n.
And now the square of nine square root 85,
we square nine which is 81.
And then we multiply by the square
of the square root of 85, which is 85.
This is equal to the square root of two n,
which is four n squared,
plus the squared of nine n, which is 81 n squared.
Combining like terms, we have 81 times 85
is equal to 85 n squared.
And I'm not going to find the product on the left
because notice now if we divide both sides by 85,
we can easily simplify the left side to just 81.
We now have 81 is equal to n squared.
And now to solve for n we take the square root
of both sides of the equation.
We are going to get a positive and negative value for n.
But because we are in the first quadrant,
we know n has to be positive.
So algebraically we have n equals plus or minus nine.
But again, because n has to be positive,
we only use n equals nine.
So if n is equal to nine, two n is equal to 18.
And nine n is equal to nine times nine, which is 81.
Which means we now know the x component is 18,
and the y component is 81.
Now you might be wondering why we didn't
just find the angle theta, using arc-tangent 9/2.
We can do this, but the angle is going to be messy,
and therefore we'd have more
of an error when determining these components.
But let's also show that.
Angle theta is equal to arc-tangent 9/2.
Going to the calculator,
in degree mode to two decimal places
we have approximately 77.47 degrees.
And now applying the formula below, where the x component
is the magnitude of v times cosine theta,
and the y component is the magnitude of v times sine theta,
we would have vector v has an x component
of nine square root 85
times cosine of 77.47 degrees.
And a y component of nine square root 85
times sine 77.47 degrees.
And now let's go to the calculator
and compare this to the exact values that we already found.
Nine square root 85 right arrow cosine 77.47 degrees.
Notice how it does not give us exactly 18.
It gives us approximately 18.0017.
And let's check the y component.
Let's go ahead and press second, enter.
And then just change cosine to sine.
And notice how the y component
would be approximately 80.9996 rather than 81.
So taking the first approach,
notice how we get the exact components.
Taking the second approach,
we only get decimal approximations.
I hope you found this helpful.