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To solve for \( |2w - v| \) with the given vectors \( v = \langle -3, 6 \rangle \) and \( w = \langle -1, -3 \rangle \), we need to follow these steps:

1. **Calculate \( 2w \):**
   \[
   2w = 2 \times \langle -1, -3 \rangle = \langle -2, -6 \rangle
   \]

2. **Calculate \( 2w - v \):**
   \[
   2w - v = \langle -2, -6 \rangle - \langle -3, 6 \rangle
   \]
   Perform the subtraction for each component:
   \[
   2w - v = \langle -2 + 3, -6 - 6 \rangle = \langle 1, -12 \rangle
   \]

3. **Find the magnitude \( |2w - v| \):**
   \[
   |2w - v| = \sqrt{(1)^2 + (-12)^2} = \sqrt{1 + 144} = \sqrt{145}
   \]

Thus, the answer is:
\[
|2w - v| = \sqrt{145}
\]

Would you like further details on any of the steps, or have additional questions?

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Here are some related questions:
1. What is the meaning of vector magnitude in this context?
2. How does multiplying a vector by a scalar affect its magnitude?
3. Can you break down vector subtraction step-by-step?
4. Why is the magnitude of a vector always positive?
5. What are some practical applications of vector calculations?

**Tip:** When working with vectors, always remember to carefully track each component during operations like addition or subtraction to avoid mistakes.

Do the indicated calculation for the vectors v = ⟨-3, 6⟩ and w = ⟨-1, -3⟩. Find |2w - v|.

Learn how to find the magnitude |2w - v| for the vectors v = ⟨-3, 6⟩ and w = ⟨-1, -3⟩. This solution covers key vector concepts, including scalar multiplication, vector subtraction, and magnitude calculation, suitable for high school students.

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