Let's solve the problem step-by-step to find the vector [tex]\(\overrightarrow{AB}\)[/tex] given points [tex]\(A, B, \)[/tex] and [tex]\(C\)[/tex].
We are provided with the coordinates of points [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex]:
- Point [tex]\(A\)[/tex] with coordinates [tex]\( (3, -5, 1) \)[/tex]
- Point [tex]\(B\)[/tex] with coordinates [tex]\( (7, 7, 2) \)[/tex]
- Point [tex]\(C\)[/tex] with coordinates [tex]\( (-1, 1, 3) \)[/tex]
To find the vector [tex]\(\overrightarrow{AB}\)[/tex], we use the formula:
[tex]\[
\overrightarrow{AB} = \overrightarrow{B} - \overrightarrow{A}
\][/tex]
Here are the coordinates for points [tex]\(B\)[/tex] and [tex]\(A\)[/tex]:
[tex]\[
\overrightarrow{B} = \begin{pmatrix} 7 \\ 7 \\ 2 \end{pmatrix}
\][/tex]
[tex]\[
\overrightarrow{A} = \begin{pmatrix} 3 \\ -5 \\ 1 \end{pmatrix}
\][/tex]
We subtract the coordinates of [tex]\(A\)[/tex] from [tex]\(B\)[/tex]:
[tex]\[
\overrightarrow{AB} = \begin{pmatrix} 7 \\ 7 \\ 2 \end{pmatrix} - \begin{pmatrix} 3 \\ -5 \\ 1 \end{pmatrix}
\][/tex]
Now, perform the subtraction component-wise:
[tex]\[
\overrightarrow{AB} = \begin{pmatrix} 7 - 3 \\ 7 - (-5) \\ 2 - 1 \end{pmatrix} = \begin{pmatrix} 4 \\ 12 \\ 1 \end{pmatrix}
\][/tex]
Therefore, the vector [tex]\(\overrightarrow{AB}\)[/tex] is:
[tex]\[
\overrightarrow{AB} = \begin{pmatrix} 4 \\ 12 \\ 1 \end{pmatrix}
\][/tex]
So, the coordinates of vector [tex]\(\overrightarrow{AB}\)[/tex] are:
[tex]\[
(4, 12, 1)
\][/tex]
