https://file.aiquickdraw.com/mathbot/file/172601603753540ryn8of.jpg

The first dropdown lists: x1, x2, x3. The second dropdown lists: basic, free

The given image shows three vectors and asks to determine the value(s) of \( h \) that make the vectors linearly dependent. The vectors are:

\[
\mathbf{v_1} = \begin{bmatrix} 3 \\ -3 \\ 6 \end{bmatrix}, \quad
\mathbf{v_2} = \begin{bmatrix} 6 \\ -9 \\ 11 \end{bmatrix}, \quad
\mathbf{v_3} = \begin{bmatrix} -4 \\ 4 \\ h \end{bmatrix}
\]

### Step-by-Step Explanation:

1. **Linear Dependence Condition:**
   The vectors will be linearly dependent if there exists a non-trivial linear combination of the vectors that results in the zero vector. Mathematically, this happens if the determinant of the matrix formed by the vectors is zero. The matrix formed by these vectors is:

   \[
   A = \begin{bmatrix} 
   3 & 6 & -4 \\ 
   -3 & -9 & 4 \\ 
   6 & 11 & h 
   \end{bmatrix}
   \]

   We need to solve for \( h \) such that the determinant of this matrix is zero.

2. **Determinant Calculation:**

   The determinant of a 3x3 matrix is calculated as follows:

   \[
   \text{det}(A) = 3 \cdot \text{det} \begin{bmatrix} -9 & 4 \\ 11 & h \end{bmatrix}
   - 6 \cdot \text{det} \begin{bmatrix} -3 & 4 \\ 6 & h \end{bmatrix}
   + (-4) \cdot \text{det} \begin{bmatrix} -3 & -9 \\ 6 & 11 \end{bmatrix}
   \]

   Let's compute these 2x2 determinants and substitute them back into the formula.

Let me calculate this determinant next.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, I can guide you through calculating the determinant manually if you'd like, or you can try using a calculator to determine it. Let me know how you'd like to proceed.

Here are five related questions you might find helpful:
1. How do we compute the determinant of a 3x3 matrix manually?
2. What is the geometric meaning of linearly dependent vectors?
3. How do we interpret the value of \( h \) in this problem?
4. What is the importance of the determinant being zero in linear dependence?
5. How does changing a parameter in one vector affect the linear dependence of a set?

**Tip:** Linear dependence implies that one of the vectors can be written as a linear combination of the others. This is why a zero determinant is critical—it indicates that the matrix (formed by vectors) is not invertible.

Find the value(s) of h for which the vectors are linearly dependent. Justify your answer.

Learn how to determine the value(s) of h that make a set of vectors linearly dependent. This problem involves key linear algebra concepts, including the calculation of a determinant for a 3x3 matrix and the condition for linear dependence (det(A) = 0). Suitable for students studying undergraduate linear algebra.

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