To find the component form of the vector [tex]\(\overrightarrow{AB}\)[/tex] given the initial point [tex]\(A(8, -2)\)[/tex] and the terminal point [tex]\(B(-2, -2)\)[/tex], we need to determine the difference between the corresponding coordinates of points B and A.
1. Identify the coordinates of points [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
- Point [tex]\(A\)[/tex] has coordinates [tex]\((8, -2)\)[/tex].
- Point [tex]\(B\)[/tex] has coordinates [tex]\((-2, -2)\)[/tex].
2. Subtract the coordinates of point [tex]\(A\)[/tex] from the coordinates of point [tex]\(B\)[/tex]:
- For the x-coordinate: [tex]\(B_x - A_x = -2 - 8 = -10\)[/tex]
- For the y-coordinate: [tex]\(B_y - A_y = -2 - (-2) = -2 + 2 = 0\)[/tex]
3. Combine these differences to form the component vector [tex]\(\overrightarrow{AB}\)[/tex]:
- So, the component form of [tex]\(\overrightarrow{AB}\)[/tex] is [tex]\(\langle -10, 0 \rangle\)[/tex].
Hence, the correct component form of the vector [tex]\(\overrightarrow{AB}\)[/tex] is [tex]\(\langle -10, 0 \rangle\)[/tex]. Therefore, the correct answer is:
[tex]\[
\langle -10, 0 \rangle
\][/tex]
