To find the component form of vector $u$ and its magnitude, let's go through each part of the problem.

1. Component Form:
   The component form of a vector $u$ with initial point $(x_1, y_1)$ and terminal point $(x_2, y_2)$ is given by: $u = \langle x_2 - x_1, y_2 - y_1 \rangle$ Here, $(x_1, y_1) = (14, -6)$ and $(x_2, y_2) = (-4, 7)$.

   $u = \langle -4 - 14, 7 - (-6) \rangle$ $u = \langle -18, 13 \rangle$

   So, the component form of $u$ is: $u = \langle -18, 13 \rangle$

2. Magnitude of $u$:
   The magnitude of $u$, denoted $||u||$, is calculated using the formula: $||u|| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Substituting in our values: $||u|| = \sqrt{(-18)^2 + 13^2}$ $||u|| = \sqrt{324 + 169}$ $||u|| = \sqrt{493} \approx 22.2$

Answer:

$u = \langle -18, 13 \rangle, \quad ||u|| \approx 22.2 \text{ units}$

Would you like a deeper explanation or have any questions?

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1. How do you calculate the direction angle of a vector?
2. What is the physical interpretation of vector magnitude?
3. How would you find the midpoint between the initial and terminal points?
4. How does vector addition work in component form?
5. Can you explain scalar multiplication for vectors?

Tip: For any vector $\langle a, b \rangle$, the direction can be found by calculating the angle $\theta = \tan^{-1}(b/a)$ with respect to the x-axis.