https://file.aiquickdraw.com/mathbot/file/1730509181033kgnxy4vi.jpg

To solve this problem, we need to calculate the magnitude (or length) of the vector \( \vec{a} = 2\vec{i} + \vec{j} - 2\vec{k} \) in 3-dimensional space.

The formula for the magnitude of a vector \( \vec{a} = a_x \vec{i} + a_y \vec{j} + a_z \vec{k} \) is:

\[
|\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}
\]

In this case:
- \( a_x = 2 \)
- \( a_y = 1 \)
- \( a_z = -2 \)

Substitute these values into the formula:

\[
|\vec{a}| = \sqrt{2^2 + 1^2 + (-2)^2}
\]

\[
|\vec{a}| = \sqrt{4 + 1 + 4}
\]

\[
|\vec{a}| = \sqrt{9}
\]

\[
|\vec{a}| = 3
\]

So, the correct answer is **3**.

Would you like more details or have any further questions?

### Related Questions
1. What is the magnitude of a vector in 2D space, and how does it differ from 3D?
2. How would the magnitude change if one component of \( \vec{a} \) were negative?
3. How is the direction of a vector related to its components?
4. Can you determine a vector's direction angle in 3D space?
5. How does vector addition affect the magnitudes of vectors?

### Tip
Remember that the magnitude of a vector is always non-negative, regardless of the signs of its components.

Trong không gian Oxyz, cho vectơ \( \vec{a} = 2\vec{i} + \vec{j} - 2\vec{k} \). Độ dài của vectơ \( \vec{a} \) bằng?

Find the magnitude of vector \( \vec{a} = 2\vec{i} + \vec{j} - 2\vec{k} \) in 3D space. This solution explains how to apply the formula for vector magnitude, covering key concepts in vector mathematics, suitable for high school students in Grades 10-12.

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