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Given that [tex]\overrightarrow{OA} = 11x + 6y[/tex], [tex]\overrightarrow{OB} = 4x + 10y[/tex], and [tex]\overrightarrow{CO} = -13x + 11y[/tex], write down each of the following vectors in its simplest form.

a) [tex]\overrightarrow{BA} = 7x - 4y \quad \square[/tex]

b) [tex]\overrightarrow{AC} = \quad \square[/tex]

Sagot :

GinnyAnswer
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]
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