Let's start by understanding the given vectors:
- [tex]\(\overrightarrow{O A} = 11x + 6y\)[/tex]
- [tex]\(\overrightarrow{O B} = 4x + 10y\)[/tex]
- [tex]\(\overrightarrow{C O} = -13x + 11y\)[/tex]
We need to find the following vectors:
a) [tex]\(\overrightarrow{B A}\)[/tex]:
The vector [tex]\(\overrightarrow{B A}\)[/tex] is obtained by subtracting vector [tex]\(\overrightarrow{O B}\)[/tex] from [tex]\(\overrightarrow{O A}\)[/tex]:
[tex]\[
\overrightarrow{B A} = \overrightarrow{O A} - \overrightarrow{O B}
\][/tex]
Substitute the given vectors:
[tex]\[
\overrightarrow{B A} = (11x + 6y) - (4x + 10y)
\][/tex]
Simplify by combining like terms:
[tex]\[
\overrightarrow{B A} = 11x + 6y - 4x - 10y
\][/tex]
[tex]\[
\overrightarrow{B A} = (11x - 4x) + (6y - 10y)
\][/tex]
[tex]\[
\overrightarrow{B A} = 7x - 4y
\][/tex]
So, [tex]\(\overrightarrow{B A} = 7x - 4y\)[/tex].
b) [tex]\(\overrightarrow{A C}\)[/tex]:
The vector [tex]\(\overrightarrow{A C}\)[/tex] is obtained by subtracting vector [tex]\(\overrightarrow{O A}\)[/tex] from [tex]\(\overrightarrow{C O}\)[/tex]. Note that [tex]\(\overrightarrow{C O}\)[/tex] points from [tex]\(C\)[/tex] to [tex]\(O\)[/tex], but we need [tex]\(\overrightarrow{O C}\)[/tex], which is the negative of [tex]\(\overrightarrow{C O}\)[/tex]:
[tex]\[
\overrightarrow{O C} = -\overrightarrow{C O} = -(-13x + 11y) = 13x - 11y
\][/tex]
Now, we can find [tex]\(\overrightarrow{A C}\)[/tex] by:
[tex]\[
\overrightarrow{A C} = \overrightarrow{O C} - \overrightarrow{O A}
\][/tex]
Substitute the given vectors:
[tex]\[
\overrightarrow{A C} = (13x - 11y) - (11x + 6y)
\][/tex]
Simplify by combining like terms:
[tex]\[
\overrightarrow{A C} = 13x - 11y - 11x - 6y
\][/tex]
[tex]\[
\overrightarrow{A C} = (13x - 11x) + (-11y - 6y)
\][/tex]
[tex]\[
\overrightarrow{A C} = 2x - 17y
\][/tex]
So, [tex]\(\overrightarrow{A C} = 2x - 17y\)[/tex].
Hence, the vectors are:
[tex]\[
\overrightarrow{B A} = 7x - 4y \quad \text{and} \quad \overrightarrow{A C} = 2x - 17y
\][/tex]
