The projections of a directed line segment on the coordinate axes are `12,4,3` respectively.
What are the direction cosines of the line segment

Options

A. `(pm(12)/(13),pm4/(13),3/(13))`
B. `((12)/(13),-4/(31),3/(13))`
C. `((12)/(13),4/(13),3/(13))`
D. `(-(12)/(13),-4/(13),3/(13))`

Answer

To find the direction cosines of a directed line segment with given projections on the coordinate axes, we can follow these steps:

### Step 1: Understand the Problem
The projections of the directed line segment on the coordinate axes are given as \(12\), \(4\), and \(3\). These correspond to the \(x\), \(y\), and \(z\) coordinates respectively.

### Step 2: Calculate the Length of the Line Segment
The length \(L\) of the line segment can be calculated using the formula:
\[ L = \sqrt{x^2 + y^2 + z^2} \]
Substituting the values:
\[ L = \sqrt{12^2 + 4^2 + 3^2} = \sqrt{144 + 16 + 9} = \sqrt{169} = 13 \]

### Step 3: Calculate the Direction Cosines
The direction cosines \(l\), \(m\), and \(n\) are defined as:
\[ l = \frac{x}{L}, \quad m = \frac{y}{L}, \quad n = \frac{z}{L} \]
Substituting the values:
- For \(l\):
\[ l = \frac{12}{13} \]
- For \(m\):
\[ m = \frac{4}{13} \]
- For \(n\):
\[ n = \frac{3}{13} \]

### Step 4: Write the Final Answer
The direction cosines of the directed line segment are:
\[ \left( \frac{12}{13}, \frac{4}{13}, \frac{3}{13} \right) \]

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