To find the correct component form of vector [tex]\( v \)[/tex], let's proceed step-by-step.
1. Determine Vector [tex]\( u \)[/tex]:
- Initial point of [tex]\( u \)[/tex]: [tex]\((15, 22)\)[/tex]
- Terminal point of [tex]\( u \)[/tex]: [tex]\((5, -4)\)[/tex]
The component form of [tex]\( u \)[/tex] is determined by subtracting 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, vector [tex]\( u \)[/tex] can be represented as:
[tex]\[
u = \langle -10, -26 \rangle
\][/tex]
2. Determine Vector [tex]\( v \)[/tex]:
We know that vector [tex]\( v \)[/tex] is twice the magnitude of [tex]\( u \)[/tex] and points in the opposite direction.
- Opposite direction: Simply negate the components of [tex]\( u \)[/tex].
- Twice the magnitude: Multiply the components by 2.
So, the components of [tex]\( v \)[/tex]:
[tex]\[
v_x = -2 \times (-10) = 20
\][/tex]
[tex]\[
v_y = -2 \times (-26) = 52
\][/tex]
Therefore, vector [tex]\( v \)[/tex] in component form is:
[tex]\[
v = \langle 20, 52 \rangle
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{D. \ v = \langle 20, 52 \rangle}
\][/tex]
