Step 1: Find the component form of the vector $\mathbf{v}$.

The component form of a vector $\mathbf{v}$ is found by subtracting the coordinates of the initial point from the coordinates of the terminal point. If the initial point is $(x_1, y_1)$ and the terminal point is $(x_2, y_2)$, the component form of the vector is:

$\mathbf{v} = \langle x_2 - x_1, y_2 - y_1 \rangle$

In this case, the initial point is $(-3, 13)$ and the terminal point is $(5, 30)$.

$\mathbf{v} = \langle 5 - (-3), 30 - 13 \rangle$

$\mathbf{v} = \langle 5 + 3, 30 - 13 \rangle = \langle 8, 17 \rangle$

Thus, the component form of the vector $\mathbf{v}$ is $\langle 8, 17 \rangle$.

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Step 2: Find the magnitude of the vector $\mathbf{v}$.

The magnitude $||\mathbf{v}||$ of a vector with component form $\langle x, y \rangle$ is calculated using the formula:

$||\mathbf{v}|| = \sqrt{x^2 + y^2}$

For the vector $\mathbf{v} = \langle 8, 17 \rangle$:

$||\mathbf{v}|| = \sqrt{8^2 + 17^2} = \sqrt{64 + 289} = \sqrt{353}$

So, the magnitude of the vector $\mathbf{v}$ is approximately:

$||\mathbf{v}|| \approx 18.79$

Final Answer:

• The component form of the vector is $\langle 8, 17 \rangle$.
• The magnitude of the vector is approximately $18.79$.

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Do you want more details or have any questions?

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Here are 5 related questions:

1. How do you find the direction angle of a vector?
2. What are the properties of unit vectors, and how can you convert a vector into a unit vector?
3. How would you calculate the dot product of two vectors?
4. What is the difference between the component form and the parametric form of a vector?
5. How can vectors be added or subtracted in terms of their components?

Tip: When calculating the magnitude of a vector, always double-check the signs of the components before squaring them to avoid errors.