The minimum distance of the point (1, 2, 3) from
x-axis is

Options

A. 1 unit
B. `sqrt(6)` units
C. `sqrt(13)` units
D. `sqrt(14)` units

Answer

To find the minimum distance of the point (1, 2, 3) from the x-axis, we can follow these steps:

### Step 1: Understand the Geometry
The x-axis can be represented by the set of points where the y-coordinate and z-coordinate are both zero. Therefore, any point on the x-axis can be represented as (x, 0, 0).

### Step 2: Identify the Coordinates
The given point is (1, 2, 3). We need to find the minimum distance from this point to the x-axis.

### Step 3: Determine the Perpendicular Point on the x-axis
The perpendicular point from (1, 2, 3) to the x-axis will have the same x-coordinate as the given point, but the y and z coordinates will be zero. Thus, the point on the x-axis is (1, 0, 0).

### Step 4: Calculate the Distance
The formula for the distance between two points in three-dimensional space, (x1, y1, z1) and (x2, y2, z2), is given by:

\[ \text{Distance} = \sqrt{(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2} \]

Applying this to our points (1, 2, 3) and (1, 0, 0):

- \(x1 = 1\), \(y1 = 2\), \(z1 = 3\)
- \(x2 = 1\), \(y2 = 0\), \(z2 = 0\)

Substituting these values into the distance formula:

\[ \text{Distance} = \sqrt{(1 - 1)^2 + (0 - 2)^2 + (0 - 3)^2} \]
\[ = \sqrt{0^2 + (-2)^2 + (-3)^2} \]
\[ = \sqrt{0 + 4 + 9} \]
\[ = \sqrt{13} \]

### Step 5: Conclusion
Thus, the minimum distance of the point (1, 2, 3) from the x-axis is \(\sqrt{13}\) units.

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