We're asked to write the given vectors
in component form using angled brackets.
This time, we will use the ordered pairs
of the initial and terminal points to determine
their component form.
Given vector v with the initial point, x sub one
comma y sub one, and the terminal point x sub two
comma y sub two, the component form of vector v
is equal to angled bracket x sub two minus x sub one
comma y sub two minus y sub one.
So if the change in x gives us the horizontal
or x component and the change in y gives us
the vertical or y component.
Let's begin by analyzing vector u.
Notice how the initial point is this point here
where the ordered pair is negative two comma negative five.
Because this is the initial point, using our formula,
this ordered pair must be x sub one comma y sub one.
And the terminal point is this point where the ordered pair
is negative six comma four.
Because this is the terminal point, this ordered pair
must be x sub two comma y sub two, which means
a component form of vector u has an x component
of x sub two minus x sub one, which is negative six
minus negative two.
And the y component is y sub two minus y sub one
which is four minus negative five.
Simplifying, the x component is negative six
minus negative two, which is equivalent to negative six
plus two, which is negative four.
And the y component is four minus negative five
which is equivalent to four plus five, which is nine.
And let's analyze the graph to verify
these are the components of vector u.
Starting at the initial point, notice how
the horizontal change or change in x is left four units
which is why the vertical component is negative four.
And then from here, the vertical change
or vertical displacement is up nine units
which is why the vertical component is positive nine.
You can also say the change in y is positive nine.
And now let's analyze vector v.
The initial point is negative four comma five.
This must be x sub one comma y sub one.
And the terminal point is here, where the ordered pair
is three comma one.
This ordered pair must be x sub two comma y sub two
which means vector v has an x component of x sub two
minus x sub one, which is three minus negative four.
And the y component is y sub two minus y sub one
which is one minus five.
Simplifying, the x component is three minus negative four
which is equivalent to three plus four, which is seven.
And the y component is one minus five
which is negative four.
Let's go ahead and verify this graphically.
Starting at the initial point, the change in x
or the horizontal change or horizontal displacement
is right one, two, three, four, five, six, seven units.
This vector does represent the horizontal or x component
which is positive seven.
And then from here, the vertical change
or vertical displacement or the change in y is down four
which is why the vertical component is negative four.
I hope you found this helpful.