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?

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.