Find the direction-cosines of the line:
`(x-1)/(2) = -y = (z + 1)/(2)`.

Answer

To find the direction cosines of the line given by the equation

\[ \frac{x-1}{2} = -y = \frac{z+1}{2}, \]

we can start by rewriting the equation in a more standard form.

### Step 1: Rewrite the equation
The equation can be separated into three parts:

1. \(\frac{x-1}{2} = -y\)
2. \(-y = \frac{z+1}{2}\)

From these, we can express \(y\) in terms of \(x\) and \(z\).

### Step 2: Express \(y\) in terms of \(x\)
From the first part, we have:

\[ y = -\frac{x-1}{2} = \frac{1-x}{2}. \]

### Step 3: Express \(z\) in terms of \(y\)
From the second part, we can express \(z\):

\[ -y = \frac{z+1}{2} \implies z + 1 = -2y \implies z = -2y - 1. \]

Substituting \(y\) from Step 2 into this equation gives:

\[ z = -2\left(\frac{1-x}{2}\right) - 1 = -(1-x) - 1 = -1 + x - 1 = x - 2. \]

### Step 4: Write the parametric equations
Now we can express \(x\), \(y\), and \(z\) in terms of a parameter \(t\):

Let \(x = 1 + 2t\). Then:

- From \(y = \frac{1-x}{2}\):

\[ y = \frac{1 - (1 + 2t)}{2} = \frac{-2t}{2} = -t. \]

- From \(z = x - 2\):

\[ z = (1 + 2t) - 2 = 2t - 1. \]

Thus, we have the parametric equations:

\[ x = 1 + 2t, \quad y = -t, \quad z = 2t - 1. \]

### Step 5: Identify direction ratios
From the parametric equations, we can identify the direction ratios of the line:

- The coefficients of \(t\) give us the direction ratios: \(2, -1, 2\).

### Step 6: Calculate direction cosines
The direction cosines \(l, m, n\) are given by the ratios of the direction ratios to the magnitude of the direction ratios. First, we calculate the magnitude:

\[ \sqrt{(2^2) + (-1^2) + (2^2)} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3. \]

Now, we can find the direction cosines:

\[ l = \frac{2}{3}, \quad m = \frac{-1}{3}, \quad n = \frac{2}{3}. \]

### Final Answer
The direction cosines of the line are:

\[ \left( \frac{2}{3}, -\frac{1}{3}, \frac{2}{3} \right). \]

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