Let x = (-1)         ( -4)        (0 )
            (0 ) , y = (-2 ), z = (1 ) The triple product x*(y*z) is:
            (9 )          (0 )         (-2)

Given the vectors \( x = \begin{pmatrix} -1 \\ 0 \\ 0 \end{pmatrix} \), \( y = \begin{pmatrix} -4 \\ -2 \\ 9 \end{pmatrix} \), and \( z = \begin{pmatrix} 0 \\ 1 \\ -2 \end{pmatrix} \), you want to calculate the scalar triple product \( x \cdot (y \times z) \).

### Step 1: Calculate the cross product \( y \times z \)
The cross product \( y \times z \) is given by:
\[
y \times z = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -4 & -2 & 9 \\ 0 & 1 & -2 \end{vmatrix}
\]

This determinant can be expanded as:
\[
y \times z = \mathbf{i} \left((-2)(-2) - (9)(1)\right) - \mathbf{j} \left((-4)(-2) - (9)(0)\right) + \mathbf{k} \left((-4)(1) - (-2)(0)\right)
\]

Simplifying this:
\[
y \times z = \mathbf{i} \left(4 - 9\right) - \mathbf{j} \left(8 - 0\right) + \mathbf{k} \left(-4 - 0\right)
\]
\[
y \times z = \begin{pmatrix} -5 \\ -8 \\ -4 \end{pmatrix}
\]

### Step 2: Calculate the dot product \( x \cdot (y \times z) \)
Now, calculate the dot product of vector \( x \) with the cross product \( y \times z \):
\[
x \cdot (y \times z) = \begin{pmatrix} -1 \\ 0 \\ 0 \end{pmatrix} \cdot \begin{pmatrix} -5 \\ -8 \\ -4 \end{pmatrix}
\]

This gives:
\[
x \cdot (y \times z) = (-1)(-5) + (0)(-8) + (0)(-4) = 5
\]

### Final Answer:
The scalar triple product \( x \cdot (y \times z) \) is **5**.

Would you like more details or have any questions about this solution? Here are some related questions:

1. What is the significance of the scalar triple product in vector algebra?
2. How do you calculate the area of a parallelogram formed by two vectors?
3. Can the scalar triple product be zero? What does that signify?
4. How is the scalar triple product used in determining the volume of a parallelepiped?
5. What are some applications of vector cross products in physics?

**Tip:** The scalar triple product \( x \cdot (y \times z) \) is a useful tool for determining the volume of the parallelepiped formed by vectors \( x \), \( y \), and \( z \).

Learn how to calculate the scalar triple product x * (y * z) using vector algebra. Step-by-step solution provided for vectors x = (-1, 0, 0), y = (-4, -2, 9), and z = (0, 1, -2). Understand the significance of the scalar triple product in vector algebra.

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