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 will follow these steps:

### Step 1: Rewrite the equation in standard form

The equation of the line is given in a form that mixes the variables. We need to express it in a standard format. We can rewrite the equation as:

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

To eliminate the negative sign in front of \(y\), we can rewrite \(-y\) as \(y/(-1)\). Thus, we have:

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

### Step 2: Identify the direction ratios

From the rewritten equation, we can extract the direction ratios. The coefficients of \(x\), \(y\), and \(z\) in the standard form are:

- For \(x\): The coefficient is \(2\),
- For \(y\): The coefficient is \(-1\),
- For \(z\): The coefficient is \(2\).

Thus, the direction ratios \( (l, m, n) \) are:

\[ l = 2, \quad m = -1, \quad n = 2. \]

### Step 3: Calculate the magnitude of the direction ratios

Next, we need to calculate the magnitude of the direction ratios to find the direction cosines. The magnitude \(R\) is given by:

\[ R = \sqrt{l^2 + m^2 + n^2} = \sqrt{2^2 + (-1)^2 + 2^2}. \]

Calculating this:

\[ R = \sqrt{4 + 1 + 4} = \sqrt{9} = 3. \]

### Step 4: Find the direction cosines

The direction cosines \( (L, M, N) \) are calculated by dividing each direction ratio by the magnitude \(R\):

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

### Final Answer

Thus, the direction cosines of the line are:

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

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