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?

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.