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

Options

A. 19 units
B. 17 units
C. 15 units
D. 13 units

Answer

To find the length of the directed line segment given its projections on the coordinate axes, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Projections**:
The projections of the directed line segment on the coordinate axes are given as \(12\), \(4\), and \(3\). We can denote these as:
- \(x = 12\) (projection on the x-axis)
- \(y = 4\) (projection on the y-axis)
- \(z = 3\) (projection on the z-axis)

2. **Use the Length Formula**:
The length \(L\) of the line segment can be calculated using the formula:
\[ L = \sqrt{x^2 + y^2 + z^2} \]

3. **Substitute the Values**:
Substitute the values of \(x\), \(y\), and \(z\) into the formula:
\[ L = \sqrt{12^2 + 4^2 + 3^2} \]

4. **Calculate the Squares**:
Calculate each square:
- \(12^2 = 144\)
- \(4^2 = 16\)
- \(3^2 = 9\)

5. **Add the Squares**:
Now, add these squared values together:
\[ 144 + 16 + 9 = 169 \]

6. **Take the Square Root**:
Finally, take the square root of the sum:
\[ L = \sqrt{169} = 13 \]

7. **Conclusion**:
Therefore, the length of the directed line segment is \(13\) units.

### Final Answer:
The length of the line segment is \(13\) units.

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