The projection of a line segment on the axis 1, 2, 3 respectively. Then find the length of line segment.

Answer

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

### Step 1: Understand the Projections
The projections of the line segment on the x, y, and z axes are given as 1, 2, and 3 respectively. This means that if we have a point \( P \) with coordinates \( (x, y, z) \), then:
- The projection on the x-axis is \( x = 1 \)
- The projection on the y-axis is \( y = 2 \)
- The projection on the z-axis is \( z = 3 \)

### Step 2: Identify the Coordinates of the Point
From the projections, we can identify the coordinates of point \( P \) as:
\[ P(1, 2, 3) \]

### Step 3: Calculate the Length of the Line Segment
The length of the line segment from the origin \( O(0, 0, 0) \) to the point \( P(1, 2, 3) \) can be calculated using the distance formula in three-dimensional space:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Substituting the coordinates of the origin and point \( P \):
\[ \text{Distance} = \sqrt{(1 - 0)^2 + (2 - 0)^2 + (3 - 0)^2} \]
\[ = \sqrt{1^2 + 2^2 + 3^2} \]
\[ = \sqrt{1 + 4 + 9} \]
\[ = \sqrt{14} \]

### Step 4: Conclusion
Thus, the length of the line segment is:
\[ \sqrt{14} \text{ units} \]

---