Find the distance between the points (-3,7,2) and (2,4,-1)

Answer

To find the distance between the points (-3, 7, 2) and (2, 4, -1), we will use the distance formula for points in three-dimensional space. The distance \(d\) between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) is given by the formula:

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

### Step 1: Identify the coordinates
Let's identify the coordinates of the two points:
- Point 1: \((x_1, y_1, z_1) = (-3, 7, 2)\)
- Point 2: \((x_2, y_2, z_2) = (2, 4, -1)\)

### Step 2: Calculate the differences in coordinates
Now, we will calculate the differences between the corresponding coordinates:
- \(x_2 - x_1 = 2 - (-3) = 2 + 3 = 5\)
- \(y_2 - y_1 = 4 - 7 = -3\)
- \(z_2 - z_1 = -1 - 2 = -3\)

### Step 3: Square the differences
Next, we will square these differences:
- \((x_2 - x_1)^2 = 5^2 = 25\)
- \((y_2 - y_1)^2 = (-3)^2 = 9\)
- \((z_2 - z_1)^2 = (-3)^2 = 9\)

### Step 4: Sum the squared differences
Now, we will sum the squared differences:
\[ 25 + 9 + 9 = 43 \]

### Step 5: Take the square root
Finally, we take the square root of the sum to find the distance:
\[ d = \sqrt{43} \]

Thus, the distance between the points (-3, 7, 2) and (2, 4, -1) is \(\sqrt{43}\).

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