Find the direction cosine of a line passing through origin and the point `(alpha, beta, gamma)`

Answer

To find the direction cosines of a line passing through the origin and the point \((\alpha, \beta, \gamma)\), we can follow these steps:

### Step 1: Identify the Points
Let the origin be point \(P(0, 0, 0)\) and the point \(Q(\alpha, \beta, \gamma)\).

### Step 2: Calculate the Distance Between the Points
The distance \(d\) between the two points \(P\) and \(Q\) can be calculated using the distance formula in three-dimensional space:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

Substituting the coordinates of points \(P\) and \(Q\):

\[ d = \sqrt{(\alpha - 0)^2 + (\beta - 0)^2 + (\gamma - 0)^2} = \sqrt{\alpha^2 + \beta^2 + \gamma^2} \]

### Step 3: Define the Direction Cosines
The direction cosines of a line are defined as the cosines of the angles that the line makes with the coordinate axes. If \(l\), \(m\), and \(n\) are the direction cosines corresponding to the x, y, and z axes respectively, they can be expressed as:

\[ l = \frac{x}{d}, \quad m = \frac{y}{d}, \quad n = \frac{z}{d} \]

Where \(x\), \(y\), and \(z\) are the coordinates of point \(Q\) and \(d\) is the distance calculated in the previous step.

### Step 4: Substitute the Values
For our point \(Q(\alpha, \beta, \gamma)\):

\[ l = \frac{\alpha}{\sqrt{\alpha^2 + \beta^2 + \gamma^2}}, \quad m = \frac{\beta}{\sqrt{\alpha^2 + \beta^2 + \gamma^2}}, \quad n = \frac{\gamma}{\sqrt{\alpha^2 + \beta^2 + \gamma^2}} \]

### Step 5: Final Result
Thus, the direction cosines of the line passing through the origin and the point \((\alpha, \beta, \gamma)\) are:

\[ l = \frac{\alpha}{\sqrt{\alpha^2 + \beta^2 + \gamma^2}}, \quad m = \frac{\beta}{\sqrt{\alpha^2 + \beta^2 + \gamma^2}}, \quad n = \frac{\gamma}{\sqrt{\alpha^2 + \beta^2 + \gamma^2}} \]