Select the correct answer.

Vector [tex]$u$[/tex] has its initial point at [tex]$(15,22)$[/tex] and its terminal point at [tex]$(5,-4)$[/tex]. Vector [tex]$v$[/tex] points in a direction opposite that of [tex]$u$[/tex], and its magnitude is twice the magnitude of [tex]$u$[/tex]. What is the component form of [tex]$v$[/tex]?

A. [tex]$v = \langle -20, 36 \rangle$[/tex]

B. [tex]$v = \langle -20, 52 \rangle$[/tex]

C. [tex]$v = \langle 20, 36 \rangle$[/tex]

D. [tex]$v = \langle 20, 52 \rangle$[/tex]



Answer :

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]