Find the coordinates of the point which is three-fifths of the way from (3, 4, 5) to (-2,-1,0).

Answer

To find the coordinates of the point which is three-fifths of the way from point A (3, 4, 5) to point B (-2, -1, 0), we can use the section formula. The section formula states that if a point divides the line segment joining two points in the ratio \( m_1 : m_2 \), then the coordinates of the point can be calculated using the following formulas:

\[ X = \frac{m_1 \cdot x_2 + m_2 \cdot x_1}{m_1 + m_2} \]
\[ Y = \frac{m_1 \cdot y_2 + m_2 \cdot y_1}{m_1 + m_2} \]
\[ Z = \frac{m_1 \cdot z_2 + m_2 \cdot z_1}{m_1 + m_2} \]

### Step-by-Step Solution:

1. **Identify the Points and Ratios:**
- Let point A be \( (x_1, y_1, z_1) = (3, 4, 5) \)
- Let point B be \( (x_2, y_2, z_2) = (-2, -1, 0) \)
- The ratio \( m_1 : m_2 \) is \( 3 : 5 \) (since we are finding three-fifths of the way).

2. **Assign Values to \( m_1 \) and \( m_2 \):**
- Here, \( m_1 = 3 \) and \( m_2 = 5 \).

3. **Calculate the X-coordinate:**
\[ X = \frac{m_1 \cdot x_2 + m_2 \cdot x_1}{m_1 + m_2} = \frac{3 \cdot (-2) + 5 \cdot 3}{3 + 5} \]
\[ = \frac{-6 + 15}{8} = \frac{9}{8} \]

4. **Calculate the Y-coordinate:**
\[ Y = \frac{m_1 \cdot y_2 + m_2 \cdot y_1}{m_1 + m_2} = \frac{3 \cdot (-1) + 5 \cdot 4}{3 + 5} \]
\[ = \frac{-3 + 20}{8} = \frac{17}{8} \]

5. **Calculate the Z-coordinate:**
\[ Z = \frac{m_1 \cdot z_2 + m_2 \cdot z_1}{m_1 + m_2} = \frac{3 \cdot 0 + 5 \cdot 5}{3 + 5} \]
\[ = \frac{0 + 25}{8} = \frac{25}{8} \]

6. **Final Coordinates:**
The coordinates of the point which is three-fifths of the way from point A to point B are:
\[ \left( \frac{9}{8}, \frac{17}{8}, \frac{25}{8} \right) \]