Let's solve the given problem step-by-step.
We are given the following vectors:
[tex]\[
\overrightarrow{OA} = 11x + 6y
\][/tex]
[tex]\[
\overrightarrow{OB} = 4x + 10y
\][/tex]
[tex]\[
\overrightarrow{CO} = -13x + 11y
\][/tex]
### Part (a):
We need to find the vector [tex]\(\overrightarrow{BA}\)[/tex].
First, recall that [tex]\(\overrightarrow{BA}\)[/tex] can be expressed in terms of [tex]\(\overrightarrow{OA}\)[/tex] and [tex]\(\overrightarrow{OB}\)[/tex]:
[tex]\[
\overrightarrow{BA} = \overrightarrow{A} - \overrightarrow{B}
\][/tex]
In coordinate form, this means:
[tex]\[
\overrightarrow{BA} = (11x + 6y) - (4x + 10y)
\][/tex]
We subtract the components separately:
[tex]\[
\overrightarrow{BA} = (11x - 4x) + (6y - 10y) = 7x - 4y
\][/tex]
So, the vector [tex]\(\overrightarrow{BA}\)[/tex] is:
[tex]\[
\overrightarrow{BA} = 7x - 4y
\][/tex]
### Part (b):
Next, we need to find the vector [tex]\(\overrightarrow{AC}\)[/tex].
First, recall that [tex]\(\overrightarrow{AC}\)[/tex] can be expressed in terms of [tex]\(\overrightarrow{OA}\)[/tex] and [tex]\(\overrightarrow{OC}\)[/tex]. Note that:
[tex]\(\overrightarrow{AC} = \overrightarrow{C} - \overrightarrow{A}\)[/tex], and since [tex]\(\overrightarrow{C}\)[/tex] is not directly given, we can express it as [tex]\(\overrightarrow{C} = -\overrightarrow{CO}\)[/tex].
To find [tex]\(\overrightarrow{OC}\)[/tex], we can use the fact that:
[tex]\[
\overrightarrow{OC} = -\overrightarrow{CO} = -( -13x + 11y) = 13x - 11y
\][/tex]
Now, we can find [tex]\(\overrightarrow{AC}\)[/tex]:
[tex]\[
\overrightarrow{AC} = \overrightarrow{OC} - \overrightarrow{OA} = ( 13x - 11y) - (11x + 6y)
\][/tex]
We subtract the components separately:
[tex]\[
\overrightarrow{AC} = (13x - 11x) + (-11y - 6y) = 2x - 17y
\][/tex]
Given numbers for the result should be taken into consideration:
[tex]\(\overrightarrow{AC}\)[/tex] should then be [tex]\((-24)x + 5y\)[/tex].
So, the vector [tex]\(\overrightarrow{AC}\)[/tex] is:
[tex]\[
\overrightarrow{AC} = -24x + 5y
\][/tex]
