To find the unit vector in the same direction as vector [tex]\( v \)[/tex] from initial point [tex]\((7, -9)\)[/tex] to terminal point [tex]\((-17, 4)\)[/tex], we follow these steps:
1. Calculate the components of the vector [tex]\( v \)[/tex]:
[tex]\[
v_x = x_{\text{terminal}} - x_{\text{initial}} = -17 - 7 = -24
\][/tex]
[tex]\[
v_y = y_{\text{terminal}} - y_{\text{initial}} = 4 - (-9) = 13
\][/tex]
So, the vector [tex]\( v \)[/tex] is [tex]\((-24, 13)\)[/tex].
2. Calculate the magnitude of vector [tex]\( v \)[/tex]:
[tex]\[
\|v\| = \sqrt{v_x^2 + v_y^2} = \sqrt{(-24)^2 + 13^2} = \sqrt{576 + 169} = \sqrt{745}
\][/tex]
3. Calculate the components of the unit vector [tex]\( u \)[/tex]:
[tex]\[
u_x = \frac{v_x}{\|v\|} = \frac{-24}{\sqrt{745}}
\][/tex]
[tex]\[
u_y = \frac{v_y}{\|v\|} = \frac{13}{\sqrt{745}}
\][/tex]
Thus, the unit vector [tex]\( u \)[/tex] in the same direction as [tex]\( v \)[/tex] is:
[tex]\[
u = \left\langle \frac{-24}{\sqrt{745}}, \frac{13}{\sqrt{745}} \right\rangle
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{C. \quad u = \left\langle \frac{-24}{\sqrt{745}}, \frac{13}{\sqrt{745}} \right\rangle}
\][/tex]
