Find the component form and the magnitude of the vector v.
Initial Point	    	Terminal Point
(−3, 13)	    	(5, 30)


### 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.

Find the component form and the magnitude of the vector v. Initial Point: (−3, 13), Terminal Point: (5, 30).

Discover how to find the component form and magnitude of a vector with initial point (−3, 13) and terminal point (5, 30). This solution involves key vector concepts and the Pythagorean Theorem. Ideal for high school students in Grades 9-12.

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