To solve the problem of finding the component form and magnitude of vector [tex]\( \mathbf{u} \)[/tex] with initial point [tex]\( P \)[/tex] at [tex]\( (0, 0) \)[/tex] and terminal point [tex]\( Q \)[/tex] at [tex]\( (5, -8) \)[/tex], we can proceed as follows:
1. Component Form:
- The component form of a vector can be determined by subtracting the coordinates of the initial point [tex]\( P \)[/tex] from the coordinates of the terminal point [tex]\( Q \)[/tex].
- The x-component of vector [tex]\( \mathbf{u} \)[/tex] is [tex]\( Q_x - P_x = 5 - 0 = 5 \)[/tex].
- The y-component of vector [tex]\( \mathbf{u} \)[/tex] is [tex]\( Q_y - P_y = -8 - 0 = -8 \)[/tex].
- Therefore, the component form of the vector [tex]\( \mathbf{u} \)[/tex] is [tex]\( \mathbf{u} = \langle 5, -8 \rangle \)[/tex].
2. Magnitude:
- The magnitude of a vector [tex]\( \mathbf{u} = \langle u_x, u_y \rangle \)[/tex] is given by [tex]\( \| \mathbf{u} \| = \sqrt{u_x^2 + u_y^2} \)[/tex].
- Substituting the components [tex]\( u_x = 5 \)[/tex] and [tex]\( u_y = -8 \)[/tex]:
[tex]\[
\| \mathbf{u} \| = \sqrt{5^2 + (-8)^2}
\][/tex]
- Simplifying inside the square root:
[tex]\[
\| \mathbf{u} \| = \sqrt{25 + 64} = \sqrt{89}
\][/tex]
Thus, the component form of vector [tex]\( \mathbf{u} \)[/tex] is [tex]\( \langle 5, -8 \rangle \)[/tex] and the magnitude of [tex]\( \mathbf{u} \)[/tex] is [tex]\( \sqrt{89} \)[/tex].
Based on the problem statement and the multiple choices provided, the correct answer is:
[tex]\[ u = \langle 5, -8 \rangle; \| u \| = \sqrt{89} \][/tex]
