The projections of a line segment on the coordinate axes are 12,4,3 respectively. The length and direction cosines of the line segment are

Options

A. `13,12/13,4/13,3/13`
B. `19,12/19,4/19,3/19`
C. `11,12/11,14/11,3/11`
D. none of these

Explanation

To solve the problem of finding the length and direction cosines of a line segment given its projections on the coordinate axes, we will follow these steps:

### Step 1: Understand the Projections
The projections of the line segment on the coordinate axes are given as:
- Projection on the x-axis: \(12\)
- Projection on the y-axis: \(4\)
- Projection on the z-axis: \(3\)

### Step 2: Represent the Line Segment
We can represent the line segment in vector form as:
\[ \vec{A} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \]
where \(a_1\), \(a_2\), and \(a_3\) are the components along the x, y, and z axes respectively.

### Step 3: Assign Values to Components
From the projections, we can directly assign:
- \(a_1 = 12\)
- \(a_2 = 4\)
- \(a_3 = 3\)

Thus, the vector representation of the line segment becomes:
\[ \vec{A} = 12 \hat{i} + 4 \hat{j} + 3 \hat{k} \]

### Step 4: Calculate the Length of the Line Segment
The length (magnitude) of the vector \(\vec{A}\) can be calculated using the formula:
\[ |\vec{A}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \]
Substituting the values:
\[ |\vec{A}| = \sqrt{12^2 + 4^2 + 3^2} = \sqrt{144 + 16 + 9} = \sqrt{169} = 13 \]

### Step 5: Calculate the Direction Cosines
The direction cosines are given by the ratios of the components of the vector to its magnitude:
\[ \text{Direction cosines} = \left( \frac{a_1}{|\vec{A}|}, \frac{a_2}{|\vec{A}|}, \frac{a_3}{|\vec{A}|} \right) \]
Substituting the values:
\[ \text{Direction cosines} = \left( \frac{12}{13}, \frac{4}{13}, \frac{3}{13} \right) \]

### Final Result
Thus, the length of the line segment is \(13\) units, and the direction cosines are:
\[ \left( \frac{12}{13}, \frac{4}{13}, \frac{3}{13} \right) \]

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