What is the distance between the points with coordinates `(-3,4,1) and (2,7,-4)`?

Options

A. 5.24
B. 7.68
C. 11.45
D. 13

Explanation

To find the distance between the points with coordinates \((-3, 4, 1)\) and \((2, 7, -4)\), we will use the distance formula for three-dimensional space.

### Step-by-Step Solution:

1. **Identify the coordinates**:
- Let the first point \( P_1 = (-3, 4, 1) \) where \( x_1 = -3 \), \( y_1 = 4 \), and \( z_1 = 1 \).
- Let the second point \( P_2 = (2, 7, -4) \) where \( x_2 = 2 \), \( y_2 = 7 \), and \( z_2 = -4 \).

2. **Write the distance formula**:
The distance \( d \) between two points in 3D space is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

3. **Substitute the coordinates into the formula**:
\[ d = \sqrt{(2 - (-3))^2 + (7 - 4)^2 + (-4 - 1)^2} \]

4. **Simplify the expressions**:
- Calculate \( x_2 - x_1 \):
\[ 2 - (-3) = 2 + 3 = 5 \]
- Calculate \( y_2 - y_1 \):
\[ 7 - 4 = 3 \]
- Calculate \( z_2 - z_1 \):
\[ -4 - 1 = -5 \]

5. **Square the differences**:
\[ d = \sqrt{(5)^2 + (3)^2 + (-5)^2} \]
- \( (5)^2 = 25 \)
- \( (3)^2 = 9 \)
- \( (-5)^2 = 25 \)

6. **Add the squares**:
\[ d = \sqrt{25 + 9 + 25} = \sqrt{59} \]

7. **Calculate the approximate value**:
The approximate value of \( \sqrt{59} \) is about \( 7.68 \).

### Final Answer:
Thus, the distance between the points \((-3, 4, 1)\) and \((2, 7, -4)\) is approximately \( 7.68 \).

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