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The question is asking about the span of the set represented by the two column vectors in \( \mathbb{R}^3 \):

\[
\left\{ \begin{pmatrix} -3 \\ 6 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix} \right\}
\]

To determine the nature of the span, let's analyze the set of vectors.

1. **Determine if the vectors are linearly independent:** 
   - The vectors are in \( \mathbb{R}^3 \), and since there are two vectors, if they are linearly independent, they will span a plane in \( \mathbb{R}^3 \).
   - To check for independence, we can see if one vector is a scalar multiple of the other. Clearly, \(\begin{pmatrix} -3 \\ 6 \\ 0 \end{pmatrix}\) is not a scalar multiple of \(\begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}\) (since their corresponding ratios are inconsistent).

2. **Conclusion about the span:**
   - Since the vectors are linearly independent, they span a plane in \( \mathbb{R}^3 \).

Thus, the correct answer is **"A plane in \( \mathbb{R}^3 \)"**.

Would you like further details or explanations on any specific point?

Here are five related questions for further exploration:
1. What defines the span of a set of vectors in general?
2. How do we determine if vectors span a line, plane, or higher-dimensional space?
3. What role does linear independence play in determining the span?
4. What is the geometric interpretation of the span in \( \mathbb{R}^2 \) versus \( \mathbb{R}^3 \)?
5. How can you express any point on the plane spanned by the given vectors?

**Tip:** For a set of vectors in \( \mathbb{R}^n \), the span can be visualized as the "shape" they form in that space (point, line, plane, etc.) depending on their linear independence and the number of vectors relative to the space's dimension.

Which of the following describe the span of the set {[-3, 6, 0], [1, 2, 0]}?

Explore the span of the set {[-3, 6, 0], [1, 2, 0]} in R^3 and determine if it forms a point, line, or plane. This problem covers key concepts in linear algebra, including vector spaces, span, and linear independence. Suitable for university-level students.

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