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To find the magnitude of vector \(\vec{w}\) with coordinates \((-11, -5)\), we can use the magnitude formula for a vector \(\vec{w} = (a, b)\):

\[
|\vec{w}| = \sqrt{a^2 + b^2}
\]

Given:
- \(a = -11\)
- \(b = -5\)

Substitute these values:

\[
|\vec{w}| = \sqrt{(-11)^2 + (-5)^2}
\]

\[
|\vec{w}| = \sqrt{121 + 25}
\]

\[
|\vec{w}| = \sqrt{146}
\]

\[
|\vec{w}| \approx 12.08
\]

Thus, the magnitude of \(\vec{w}\), rounded to the nearest hundredth, is approximately **12.08**.

Would you like further details or have any questions?

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Here are five related questions for practice:
1. What is the magnitude of a vector with coordinates \((3, 4)\)?
2. If a vector has a magnitude of \(10\), what are possible values for its components?
3. How would the magnitude formula change in three dimensions?
4. What is the direction (angle) of vector \(\vec{w}\) with respect to the x-axis?
5. How do negative signs in the components affect the magnitude of a vector?

**Tip:** Remember, the magnitude of a vector is always non-negative because it represents a distance in space.

What is the magnitude of vector w with coordinates (-11, -5)?

Learn how to calculate the magnitude of a vector with coordinates (-11, -5) using the distance formula. This problem involves vector concepts and the Pythagorean theorem, making it ideal for high school students in Grades 9-12.

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