Find the distance of the point (1,2,3) from the cor-ordinate axes.

Answer

To find the distance of the point \( A(1, 2, 3) \) from the coordinate axes, we will calculate the distance from the x-axis, y-axis, and z-axis separately.

### Step 1: Distance from the x-axis
The coordinates on the x-axis can be represented as \( (x, 0, 0) \). For our point \( A(1, 2, 3) \), we can consider the point on the x-axis that corresponds to the x-coordinate of point A, which is \( (1, 0, 0) \).

Using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
where \( (x_1, y_1, z_1) = (1, 2, 3) \) and \( (x_2, y_2, z_2) = (1, 0, 0) \).

Substituting the values:
\[ d = \sqrt{(1 - 1)^2 + (2 - 0)^2 + (3 - 0)^2} \]
\[ = \sqrt{0^2 + 2^2 + 3^2} \]
\[ = \sqrt{0 + 4 + 9} \]
\[ = \sqrt{13} \]

### Step 2: Distance from the y-axis
The coordinates on the y-axis can be represented as \( (0, y, 0) \). For our point \( A(1, 2, 3) \), we can consider the point on the y-axis that corresponds to the y-coordinate of point A, which is \( (0, 2, 0) \).

Using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
where \( (x_1, y_1, z_1) = (1, 2, 3) \) and \( (x_2, y_2, z_2) = (0, 2, 0) \).

Substituting the values:
\[ d = \sqrt{(0 - 1)^2 + (2 - 2)^2 + (0 - 3)^2} \]
\[ = \sqrt{(-1)^2 + 0^2 + (-3)^2} \]
\[ = \sqrt{1 + 0 + 9} \]
\[ = \sqrt{10} \]

### Step 3: Distance from the z-axis
The coordinates on the z-axis can be represented as \( (0, 0, z) \). For our point \( A(1, 2, 3) \), we can consider the point on the z-axis that corresponds to the z-coordinate of point A, which is \( (0, 0, 3) \).

Using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
where \( (x_1, y_1, z_1) = (1, 2, 3) \) and \( (x_2, y_2, z_2) = (0, 0, 3) \).

Substituting the values:
\[ d = \sqrt{(0 - 1)^2 + (0 - 2)^2 + (3 - 3)^2} \]
\[ = \sqrt{(-1)^2 + (-2)^2 + 0^2} \]
\[ = \sqrt{1 + 4 + 0} \]
\[ = \sqrt{5} \]

### Summary of Distances
- Distance from the x-axis: \( \sqrt{13} \)
- Distance from the y-axis: \( \sqrt{10} \)
- Distance from the z-axis: \( \sqrt{5} \)