We are given two points, \(A = (3, 6, -1)\) and \(B = (6, 2, -2)\). The point \(P\) lies on the line segment \(AB\), and its \(y\)-coordinate is \(4\). We need to find its \(z\)-coordinate.
Let \(P\) divide the line segment \(AB\) in the ratio \(\lambda:1\).
Step 1: Find the Ratio (\(\lambda\))
We use the section formula for the \(y\)-coordinate, where \(y=4\), \(y_1=6\), and \(y_2=2\):
\[y = \frac{\lambda y_2 + y_1}{\lambda + 1}\]
\[4 = \frac{\lambda(2) + 6}{\lambda + 1}\]
\[4(\lambda + 1) = 2\lambda + 6\]
\[4\lambda + 4 = 2\lambda + 6\]
\[2\lambda = 2\]
\[\mathbf{\lambda = 1}\]
The point \(P\) is the **midpoint** of the segment \(AB\) since \(\lambda = 1\).
Step 2: Find the \(z\)-coordinate (\(z\))
Now, we use the section formula for the \(z\)-coordinate with \(\lambda=1\), \(z_1=-1\), and \(z_2=-2\):
\[z = \frac{\lambda z_2 + z_1}{\lambda + 1}\]
\[z = \frac{(1)(-2) + (-1)}{1 + 1}\]
\[z = \frac{-2 - 1}{2}\]
\[\mathbf{z = -\frac{3}{2}}\]