Let's solve the question step by step.
1. Find the components of vector [tex]\( u \)[/tex]:
The initial point of [tex]\( u \)[/tex] is [tex]\( (15, 22) \)[/tex], and the terminal point is [tex]\( (5, -4) \)[/tex]. To find the component form of vector [tex]\( u \)[/tex], we subtract the coordinates of the initial point from the coordinates of the terminal point:
[tex]\[
u_x = 5 - 15 = -10
\][/tex]
[tex]\[
u_y = -4 - 22 = -26
\][/tex]
Therefore, the component form of vector [tex]\( u \)[/tex] is:
[tex]\[
u = \langle -10, -26 \rangle
\][/tex]
2. Determine the magnitude of vector [tex]\( u \)[/tex]:
The magnitude of vector [tex]\( u \)[/tex] is given by the formula:
[tex]\[
|u| = \sqrt{(u_x)^2 + (u_y)^2}
\][/tex]
Plugging in the components of [tex]\( u \)[/tex]:
[tex]\[
|u| = \sqrt{(-10)^2 + (-26)^2} = \sqrt{100 + 676} = \sqrt{776} \approx 27.85677655436824
\][/tex]
3. Find the components of vector [tex]\( v \)[/tex]:
Vector [tex]\( v \)[/tex] points in the direction opposite to vector [tex]\( u \)[/tex], and its magnitude is twice that of [tex]\( u \)[/tex]. Since [tex]\( v \)[/tex] points in the opposite direction, we need to negate the components of [tex]\( u \)[/tex] and multiply by 2:
[tex]\[
v_x = -2 \times (-10) = 20
\][/tex]
[tex]\[
v_y = -2 \times (-26) = 52
\][/tex]
Therefore, the component form of vector [tex]\( v \)[/tex] is:
[tex]\[
v = \langle 20, 52 \rangle
\][/tex]
The correct answer is:
[tex]\[
\boxed{\langle 20, 52 \rangle}
\][/tex]
So, the correct answer is:
D. [tex]\( v = \langle 20, 52 \rangle \)[/tex].
