Find the equation of line passing through (2, -1, 3) and equally inclined to the axes .

Answer

To find the equation of the line passing through the point (2, -1, 3) and equally inclined to the axes, we can follow these steps:

### Step 1: Understand the concept of inclination
A line that is equally inclined to the axes means that the angles it makes with the x, y, and z axes are the same. Let's denote these angles as \( \alpha \), \( \beta \), and \( \gamma \).

### Step 2: Establish the relationship between direction cosines
The direction cosines \( L \), \( M \), and \( N \) of the line are given by:
- \( L = \cos(\alpha) \)
- \( M = \cos(\beta) \)
- \( N = \cos(\gamma) \)

Since the line is equally inclined to the axes, we have:
\[ \alpha = \beta = \gamma \]

### Step 3: Use the property of direction cosines
From the property of direction cosines, we know:
\[ L^2 + M^2 + N^2 = 1 \]

Substituting \( L = M = N \) (since they are equal), we get:
\[ 3L^2 = 1 \]
Thus,
\[ L^2 = \frac{1}{3} \]
This gives us:
\[ L = M = N = \frac{1}{\sqrt{3}} \]

### Step 4: Write the equation of the line
The equation of a line in space that passes through a point \( (x_1, y_1, z_1) \) with direction ratios \( (L, M, N) \) is given by:
\[ \frac{x - x_1}{L} = \frac{y - y_1}{M} = \frac{z - z_1}{N} \]

Substituting the point \( (2, -1, 3) \) and the direction cosines \( L = M = N = \frac{1}{\sqrt{3}} \), we have:
\[ \frac{x - 2}{\frac{1}{\sqrt{3}}} = \frac{y + 1}{\frac{1}{\sqrt{3}}} = \frac{z - 3}{\frac{1}{\sqrt{3}}} \]

### Step 5: Simplify the equation
Multiplying through by \( \sqrt{3} \) to eliminate the fractions gives:
\[ \sqrt{3}(x - 2) = \sqrt{3}(y + 1) = \sqrt{3}(z - 3) \]

Thus, the final equation of the line is:
\[ \sqrt{3}(x - 2) = \sqrt{3}(y + 1) = \sqrt{3}(z - 3) \]

### Summary of the solution
The equation of the line passing through the point (2, -1, 3) and equally inclined to the axes is:
\[ \sqrt{3}(x - 2) = \sqrt{3}(y + 1) = \sqrt{3}(z - 3) \]

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