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

Answer

To find the direction cosines of the line given by the equation \((x-1)/(2) = -y = (z+1)/(2)\), we can follow these steps:

### Step-by-Step Solution:

1. **Rewrite the Equation**:
The equation can be rewritten in a more manageable form. We have:
\[ \frac{x - 1}{2} = -y = \frac{z + 1}{2} \]
Let’s denote this common ratio as \(k\):
\[ \frac{x - 1}{2} = k, \quad -y = k, \quad \frac{z + 1}{2} = k \]

2. **Express Variables in Terms of \(k\)**:
From the equations, we can express \(x\), \(y\), and \(z\) in terms of \(k\):
\[ x = 2k + 1 \]
\[ y = -k \]
\[ z = 2k - 1 \]

3. **Identify Direction Ratios**:
The direction ratios of the line can be obtained from the coefficients of \(k\) in the expressions for \(x\), \(y\), and \(z\). Thus, we have:
- For \(x\): The coefficient is \(2\).
- For \(y\): The coefficient is \(-1\).
- For \(z\): The coefficient is \(2\).

Therefore, the direction ratios \((a, b, c)\) are:
\[ (2, -1, 2) \]

4. **Calculate the Magnitude of the Direction Ratios**:
The magnitude of the direction ratios is given by:
\[ \sqrt{a^2 + b^2 + c^2} = \sqrt{2^2 + (-1)^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \]

5. **Find the Direction Cosines**:
The direction cosines \((l, m, n)\) are obtained by dividing each direction ratio by the magnitude:
\[ l = \frac{a}{\sqrt{a^2 + b^2 + c^2}} = \frac{2}{3}, \quad m = \frac{b}{\sqrt{a^2 + b^2 + c^2}} = \frac{-1}{3}, \quad n = \frac{c}{\sqrt{a^2 + b^2 + c^2}} = \frac{2}{3} \]

Therefore, the direction cosines are:
\[ \left(\frac{2}{3}, -\frac{1}{3}, \frac{2}{3}\right) \]

### Final Answer:
The direction cosines of the line are \(\frac{2}{3}, -\frac{1}{3}, \frac{2}{3}\).

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