Find the distance between each of pair of point:
`(2,5,3 and (-3,2,1),`

Answer

To find the distance between the points \( A(2, 5, 3) \) and \( B(-3, 2, 1) \), we will use the distance formula for points in three-dimensional space. The formula 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
Let:
- Point \( A \) have coordinates \( (x_1, y_1, z_1) = (2, 5, 3) \)
- Point \( B \) have coordinates \( (x_2, y_2, z_2) = (-3, 2, 1) \)

### Step 2: Substitute the coordinates into the formula
Now, we will substitute the coordinates into the distance formula:

\[ d = \sqrt{((-3) - 2)^2 + (2 - 5)^2 + (1 - 3)^2} \]

### Step 3: Calculate each component
Now, we will calculate each component step by step:

1. Calculate \( x_2 - x_1 \):
\[ -3 - 2 = -5 \]
Therefore, \( (-5)^2 = 25 \).

2. Calculate \( y_2 - y_1 \):
\[ 2 - 5 = -3 \]
Therefore, \( (-3)^2 = 9 \).

3. Calculate \( z_2 - z_1 \):
\[ 1 - 3 = -2 \]
Therefore, \( (-2)^2 = 4 \).

### Step 4: Sum the squares
Now, we will sum these squared values:
\[ 25 + 9 + 4 = 38 \]

### Step 5: Take the square root
Finally, we take the square root to find the distance:
\[ d = \sqrt{38} \]

### Conclusion
Thus, the distance between the points \( A(2, 5, 3) \) and \( B(-3, 2, 1) \) is:
\[ d = \sqrt{38} \text{ units} \]

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