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The problem is asking for the value(s) of \( h \) such that the vector \( \mathbf{b} = \begin{bmatrix} 4 \\ 1 \\ h \end{bmatrix} \) lies in the plane spanned by the vectors \( \mathbf{a}_1 = \begin{bmatrix} 1 \\ 4 \\ -2 \end{bmatrix} \) and \( \mathbf{a}_2 = \begin{bmatrix} -2 \\ -3 \\ 7 \end{bmatrix} \).

For \( \mathbf{b} \) to be in the plane spanned by \( \mathbf{a}_1 \) and \( \mathbf{a}_2 \), there must exist scalars \( x_1 \) and \( x_2 \) such that:

\[
x_1 \mathbf{a}_1 + x_2 \mathbf{a}_2 = \mathbf{b}
\]

This can be written as a system of equations by expressing the vectors explicitly:

\[
x_1 \begin{bmatrix} 1 \\ 4 \\ -2 \end{bmatrix} + x_2 \begin{bmatrix} -2 \\ -3 \\ 7 \end{bmatrix} = \begin{bmatrix} 4 \\ 1 \\ h \end{bmatrix}
\]

Breaking this into component equations, we get:

1. \( x_1 - 2x_2 = 4 \)
2. \( 4x_1 - 3x_2 = 1 \)
3. \( -2x_1 + 7x_2 = h \)

We can solve the first two equations for \( x_1 \) and \( x_2 \), and then substitute those values into the third equation to solve for \( h \).

### Solving the system:

1. From equation 1: \( x_1 = 4 + 2x_2 \)
2. Substitute \( x_1 = 4 + 2x_2 \) into equation 2:

\[
4(4 + 2x_2) - 3x_2 = 1
\]

\[
16 + 8x_2 - 3x_2 = 1
\]

\[
16 + 5x_2 = 1
\]

\[
5x_2 = -15
\]

\[
x_2 = -3
\]

3. Substitute \( x_2 = -3 \) into \( x_1 = 4 + 2x_2 \):

\[
x_1 = 4 + 2(-3) = 4 - 6 = -2
\]

Now that we have \( x_1 = -2 \) and \( x_2 = -3 \), substitute these into the third equation \( -2x_1 + 7x_2 = h \):

\[
-2(-2) + 7(-3) = h
\]

\[
4 - 21 = h
\]

\[
h = -17
\]

Thus, the value of \( h \) is \( \boxed{-17} \).

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Would you like a detailed explanation of any step or more examples? Here are 5 related questions you might find interesting:

1. What is the geometric interpretation of a vector being in the span of two other vectors?
2. How do we solve systems of linear equations using matrices?
3. Can vectors in \(\mathbb{R}^3\) span more than a plane?
4. How can we check if a set of vectors are linearly independent?
5. What does it mean if the system of equations has no solution?

**Tip**: When dealing with systems of linear equations, using matrix methods like Gaussian elimination can help solve them efficiently.

For what value(s) of h is b in the plane spanned by a1 and a2, where a1 = [1, 4, -2], a2 = [-2, -3, 7], and b = [4, 1, h]?

This problem explores how to determine the value of h for which the vector b lies in the plane spanned by two other vectors a1 and a2. The solution involves solving a system of linear equations derived from the linear combination of vectors. The process uses concepts from Linear Algebra, including vector spaces and spans.

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