Find the component form of vector v
if the magnitude of vector v equals 5.5
and when drawn in the standard position,
vector v lies along the negative y-axis.
And therefore this would be the vector v
where the magnitude or length is 5.5 units.
The direction of vector v
is the angle in standard position
where the initial side is along the positive x-axis.
and the terminal side is along the vector
and therefore the angle theta,
the direction of the vector
is this angle here,
which measures 270 degrees.
Now we're asked to find the component form of this vector,
and because the vector lies along the negative y-axis,
we should not have to perform any calculations
to determine the component form.
Because there is no horizontal change
or horizontal displacement,
the x component is zero,
indicates the vector points down along the negative y-axis,
the vertical component is negative 5.5
because the vertical displacement or vertical change
is down 5.5 units.
However, if we wanted to,
we could still find the x and y components
by determining the magnitude of vector v time cosine theta
and the magnitude of v times sine theta,
using the magnitude and the angle of 270 degrees.
So let's show this as well.
The vector v has an x component of
5.5 times cosine 270 degrees
and a y component of
5.5 times sine 270 degrees.
Now because the angle is a quadrantal angle,
we would need to use the unit circle
to determine the trig function values
or analyze the graph of sine and cosine.
Looking at the unit circle,
the terminal side of 270 degrees intersects the unit circle
at this point here
where the unit circle x equals cosine theta
and y equals sine theta
and therefore cosine 270 degrees is equal to zero.
Sine 270 degrees is equal to negative one.
Which gives us an x component of 5.5 times zero
and a y component of 5.5 times negative one,
of course giving us the same result
where the x component is zero
and the y component is negative 5.5.
Let's take a look at a second example.
We're asked to find the component form of the vector v
if the magnitude is seven
and when drawn in the standard position,
vector v lies along the positive x-axis.
So this would be vector v
lying along the positive x-axis,
where the length or magnitude is seven units.
And because there is no rotation from the positive x-axis,
the direction or the angle theta is zero degrees.
So once again,
to find the component form of this vector,
we should not have to perform any calculations,
because the vector v lies along the positive x-axis.
The horizontal change or displacement
is seven units to the right
and therefore the x component is seven
and there is no vertical change or vertical displacement
and therefore the vertical component,
the y component is zero.
But again if they wanted to,
we could go through the process, once again,
of determining the magnitude of v times cosine theta
and the magnitude of v times sine theta.
So once again let's also show this.
Vector v has an x component of
seven times cosine zero degrees
and a y component of seven times sine zero degrees.
Again, we have a quadrantal angle
so we will go back to the unit circle
to determine the trig function values.
The terminal side of zero degrees
intersects the unit circle at this point
and therefore cosine zero degrees
is equal to one,
sine zero degrees is equal to zero,
giving it an x component of
seven times one
and a y component of seven times zero,
of course giving us the same result
where the x component is seven
and the y component is zero.
I hope you found this helpful.