### Instruction: The point $(1,1,1)$ is rotated $180^\circ$ about the $y$-axis, then reflected through the $yz$-plane, reflected through the $xz$-plane, rotated $180^\circ$ about the $y$-axis, and reflected through the $xz$-plane.  Find the coordinates of the point now.### Response: I need to keep track of how the point changes after each transformation.
I'll use a matrix to represent the point as a column vector, and use matrix multiplication to apply the transformations.
The point $(1,1,1)$ can be written as $\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$.
To rotate $180^\circ$ about the $y$-axis, I need to multiply by the matrix $\begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$.
This gives me $\begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} -1 \\ 1 \\ -1 \end{bmatrix}$.
So the point after the first rotation is $(-1,1,-1)$.
To reflect through the $yz$-plane, I need to multiply by the matrix $\begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$.
This gives me $\begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} -1 \\ 1 \\ -1 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix}$.
So the point after the first reflection is $(1,1,-1)$.
To reflect through the $xz$-plane, I need to multiply by the matrix $\begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$.
This gives me $\begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix} = \begin{bmatrix} 1 \\ -1 \\ -1 \end{bmatrix}$.
So the point after the second reflection is $(1,-1,-1)$.
To rotate $180^\circ$ about the $y$-axis again, I need to multiply by the matrix $\begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$.
This gives me $\begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix} \begin{bmatrix} 1 \\ -1 \\ -1 \end{bmatrix} = \begin{bmatrix} -1 \\ -1 \\ 1 \end{bmatrix}$.
So the point after the second rotation is $(-1,-1,1)$.
To reflect through the $xz$-plane again, I need to multiply by the matrix $\begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$.
This gives me $\begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} -1 \\ -1 \\ 1 \end{bmatrix} = \begin{bmatrix} -1 \\ 1 \\ 1 \end{bmatrix}$.
So the point after the final reflection is $(-1,1,1)$.
These are the coordinates of the point now.