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 $z$.