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Find the component form and the magnitude of the vector v.

We are given a vector **v** with initial point (-1, -1) and terminal point (4, 5). Let's find both the component form of the vector and its magnitude.

### 1. **Component Form of the Vector**:

The component form of a vector **v** is found by subtracting the coordinates of the initial point from the coordinates of the terminal point:

\[
\mathbf{v} = (x_2 - x_1, y_2 - y_1)
\]

Where:
- \((x_1, y_1)\) is the initial point \((-1, -1)\),
- \((x_2, y_2)\) is the terminal point \((4, 5)\).

So, the components are:
\[
\mathbf{v} = (4 - (-1), 5 - (-1)) = (4 + 1, 5 + 1) = (5, 6)
\]

Thus, the component form of the vector **v** is:
\[
\mathbf{v} = (5, 6)
\]

---

### 2. **Magnitude of the Vector**:

The magnitude (or length) of the vector **v** is given by the formula:

\[
|\mathbf{v}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting the values:

\[
|\mathbf{v}| = \sqrt{(5)^2 + (6)^2} = \sqrt{25 + 36} = \sqrt{61}
\]

So, the magnitude of the vector **v** is approximately:
\[
|\mathbf{v}| \approx 7.81
\]

---

### Final Answers:
1. **Component form**: \((5, 6)\)
2. **Magnitude**: \(\sqrt{61} \approx 7.81\)

Would you like more details on these calculations or have any further questions?

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Here are 5 relative questions to expand on this:
1. How do you find the direction of a vector given its components?
2. What is the significance of a vector's magnitude in physics?
3. How can you add two vectors if you know their component forms?
4. What is the dot product of two vectors, and how is it computed?
5. How do you compute the angle between two vectors?

**Tip**: When finding the magnitude of a vector, it's like applying the Pythagorean theorem to the difference between the vector's components.

Learn how to find the component form and magnitude of a vector given its initial and terminal points. This step-by-step solution involves the key concepts of vector algebra and the Pythagorean theorem. Suitable for students in Grades 9-11.

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