Find the distance between the points `(k, k+1, k+2) and (0, 1, 2).`

Answer

To find the distance between the points \( A(k, k+1, k+2) \) and \( B(0, 1, 2) \), we can use the distance formula in three-dimensional space. The distance \( d \) between two points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

### Step 1: Identify the coordinates of the points
We have:
- Point \( A \) has coordinates \( (k, k+1, k+2) \)
- Point \( B \) has coordinates \( (0, 1, 2) \)

### Step 2: Substitute the coordinates into the distance formula
Substituting the coordinates into the distance formula, we get:

\[ d = \sqrt{(0 - k)^2 + (1 - (k + 1))^2 + (2 - (k + 2))^2} \]

### Step 3: Simplify each term
Now, simplify each term inside the square root:

1. For the first term:
\[ (0 - k)^2 = k^2 \]

2. For the second term:
\[ 1 - (k + 1) = 1 - k - 1 = -k \quad \Rightarrow \quad (-k)^2 = k^2 \]

3. For the third term:
\[ 2 - (k + 2) = 2 - k - 2 = -k \quad \Rightarrow \quad (-k)^2 = k^2 \]

### Step 4: Combine the terms
Now, substituting these simplified terms back into the distance formula:

\[ d = \sqrt{k^2 + k^2 + k^2} = \sqrt{3k^2} \]

### Step 5: Factor out the square root
We can further simplify this:

\[ d = \sqrt{3} \cdot \sqrt{k^2} = \sqrt{3} \cdot |k| \]

### Final Result
Thus, the distance between the points \( (k, k+1, k+2) \) and \( (0, 1, 2) \) is:

\[ d = \sqrt{3} |k| \]