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Given points [tex]\( A(3, -5, 1), B(7, 7, 2), C(-1, 1, 3) \)[/tex]:

(a) Find the vector [tex]\(\overrightarrow{AB}\)[/tex].

[tex]\[
\begin{aligned}
\overrightarrow{AB} &= \overrightarrow{OB} - \overrightarrow{OA} \\
&= \left( \begin{array}{c}
7 \\
7 \\
2
\end{array} \right) - \left( \begin{array}{c}
3 \\
-5 \\
1
\end{array} \right) \\
&= \left( \begin{array}{c}
4 \\
12 \\
1
\end{array} \right)
\end{aligned}
\][/tex]

Sagot :

GinnyAnsweridn
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]
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