The equations of the line QR are where `Q (3, 6, 3) , R(18, 43, 13)`

Options

A. `(x-12)/15= (y-22)/37=( z-4)/10`
B. `(x-3)/3= (y-15)/7= (z-6)/2`
C. `(x-3)/15= (y-6)/37= (z-3)/10`
D. `(x+3)/3= (y+6)/7= (z+3)/3`

Answer

To find the equation of the line QR in 3D space, we can use the formula for the equation of a line given two points. The points Q and R are given as:

- Q(3, 6, 3)
- R(18, 43, 13)

### Step-by-Step Solution:

1. **Identify the coordinates of points Q and R:**
- Point Q has coordinates (x1, y1, z1) = (3, 6, 3)
- Point R has coordinates (x2, y2, z2) = (18, 43, 13)

2. **Use the formula for the equation of a line in 3D:**
The equation of the line passing through two points (x1, y1, z1) and (x2, y2, z2) can be expressed as:
\[ \frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1} \]

3. **Substitute the coordinates into the formula:**
- For x: \( x_1 = 3 \) and \( x_2 = 18 \)
- For y: \( y_1 = 6 \) and \( y_2 = 43 \)
- For z: \( z_1 = 3 \) and \( z_2 = 13 \)

Plugging these values into the formula gives:
\[ \frac{x - 3}{18 - 3} = \frac{y - 6}{43 - 6} = \frac{z - 3}{13 - 3} \]

4. **Calculate the differences:**
- \( 18 - 3 = 15 \)
- \( 43 - 6 = 37 \)
- \( 13 - 3 = 10 \)

Thus, the equation becomes:
\[ \frac{x - 3}{15} = \frac{y - 6}{37} = \frac{z - 3}{10} \]

5. **Final equation of the line:**
The equation of the line QR can be expressed as:
\[ \frac{x - 3}{15} = \frac{y - 6}{37} = \frac{z - 3}{10} \]