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This image presents several vector magnitude calculations in different dimensions. Let's go through each one step by step to verify the results.

### (a) **Vector** \( \mathbf{v} = \langle 1, -1 \rangle \)
- The magnitude of a 2D vector \( \mathbf{v} = \langle v_1, v_2 \rangle \) is calculated as:
  \[
  \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2}
  \]
  For \( \mathbf{v} = \langle 1, -1 \rangle \):
  \[
  \|\mathbf{v}\| = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}
  \]
  **Answer is correct**: \( \|\mathbf{v}\| = \sqrt{2} \).

### (b) **Vector** \( \mathbf{v} = -9\hat{i} + 3\hat{j} \)
- This is also a 2D vector \( \mathbf{v} = \langle -9, 3 \rangle \). The magnitude is:
  \[
  \|\mathbf{v}\| = \sqrt{(-9)^2 + 3^2} = \sqrt{81 + 9} = \sqrt{90} = 3\sqrt{10}
  \]
  **Answer is correct**: \( \|\mathbf{v}\| = 3\sqrt{10} \).

### (c) **Vector** \( \mathbf{v} = \langle -1, 3, 2 \rangle \)
- For a 3D vector \( \mathbf{v} = \langle v_1, v_2, v_3 \rangle \), the magnitude is:
  \[
  \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2}
  \]
  For \( \mathbf{v} = \langle -1, 3, 2 \rangle \):
  \[
  \|\mathbf{v}\| = \sqrt{(-1)^2 + 3^2 + 2^2} = \sqrt{1 + 9 + 4} = \sqrt{14}
  \]
  **Answer is correct**: \( \|\mathbf{v}\| = \sqrt{14} \).

### (d) **Vector** \( \mathbf{v} = -\hat{i} + 2\hat{j} + 3\hat{k} \)
- This is another 3D vector. We calculate its magnitude similarly:
  \[
  \|\mathbf{v}\| = \sqrt{(-1)^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14}
  \]
  **Answer is correct**: \( \|\mathbf{v}\| = \sqrt{14} \).

All answers provided in the image are correct!

Do you need further explanations or have any specific questions?

Here are 5 related questions to extend this topic:
1. How would you calculate the unit vector for each of these vectors?
2. What happens to the magnitude of a vector if you multiply it by a scalar?
3. Can you compute the magnitude of a vector in higher dimensions, say 4D or 5D?
4. How does the magnitude formula change when dealing with complex numbers?
5. What is the physical interpretation of the magnitude of a vector in physics?

**Tip**: Always double-check signs and square each component carefully when calculating magnitudes!

Calculate the magnitude of the vectors given in the image.

Learn how to calculate the magnitude of vectors in 2D and 3D space using the Pythagorean theorem. Detailed solutions for vectors like v = <1, -1> and v = -i + 2j + 3k. Suitable for high school students in Grades 9-12.

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