To solve this problem, let's follow these steps:
1. Determine the component form of 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 found 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]
- Thus, the component form of [tex]\( u \)[/tex] is:
[tex]\[
u = \langle -10, -26 \rangle
\][/tex]
2. Calculate the magnitude of vector [tex]\( u \)[/tex]:
- The magnitude [tex]\( |u| \)[/tex] of vector [tex]\( u \)[/tex] is given by the formula:
[tex]\[
|u| = \sqrt{u_x^2 + u_y^2}
\][/tex]
- Substituting [tex]\( u_x = -10 \)[/tex] and [tex]\( u_y = -26 \)[/tex]:
[tex]\[
|u| = \sqrt{(-10)^2 + (-26)^2} = \sqrt{100 + 676} = \sqrt{776}
\][/tex]
3. Determine the magnitude of vector [tex]\( v \)[/tex]:
- The magnitude of vector [tex]\( v \)[/tex] is twice the magnitude of [tex]\( u \)[/tex]:
[tex]\[
|v| = 2 \times |u| = 2 \times \sqrt{776} = \sqrt{4 \times 776} = \sqrt{3104}
\][/tex]
- Therefore, the magnitude of [tex]\( v \)[/tex] is [tex]\( \sqrt{3104} \)[/tex].
4. Determine the direction of vector [tex]\( v \)[/tex]:
- Vector [tex]\( v \)[/tex] points in the opposite direction to vector [tex]\( u \)[/tex]. Therefore, the components of [tex]\( v \)[/tex] will be negative of the components of [tex]\( u \)[/tex], and since [tex]\( v \)[/tex] has twice the magnitude of [tex]\( u \)[/tex], each component of [tex]\( v \)[/tex] will be twice the negative components of [tex]\( u \)[/tex]:
[tex]\[
v_x = -2 \times (-10) = 20
\][/tex]
[tex]\[
v_y = -2 \times (-26) = 52
\][/tex]
5. Write the component form of vector [tex]\( v \)[/tex]:
- The component form of [tex]\( v \)[/tex] is:
[tex]\[
v = \langle 20, 52 \rangle
\][/tex]
Therefore, the correct answer is:
D. [tex]\( v = (20, 52) \)[/tex]
