To find the equation of the line passing through the points \( P(-1, 3, 2) \) and \( Q(-4, 2, -2) \), and to determine the value of \( \lambda \) for the point \( R(5, 5, \lambda) \) being collinear with \( P \) and \( Q \), we can follow these steps:
### Step 1: Determine the direction ratios of the line
The direction ratios of the line passing through points \( P \) and \( Q \) can be found using the coordinates of these points.
Given:
- \( P(x_1, y_1, z_1) = (-1, 3, 2) \)
- \( Q(x_2, y_2, z_2) = (-4, 2, -2) \)
The direction ratios \( (a, b, c) \) can be calculated as:
\[
a = x_2 - x_1 = -4 - (-1) = -4 + 1 = -3
\]
\[
b = y_2 - y_1 = 2 - 3 = -1
\]
\[
c = z_2 - z_1 = -2 - 2 = -4
\]
Thus, the direction ratios are \( (-3, -1, -4) \).
### Step 2: Write the parametric equations of the line
Using the point-direction form of the line, we can write the parametric equations as:
\[
\frac{x + 1}{-3} = \frac{y - 3}{-1} = \frac{z - 2}{-4}
\]
### Step 3: Find the value of \( \lambda \) for point \( R \)
The point \( R(5, 5, \lambda) \) is collinear with points \( P \) and \( Q \). This means that the coordinates of \( R \) must satisfy the line equation derived above.
Substituting \( R(5, 5, \lambda) \) into the parametric equations:
\[
\frac{5 + 1}{-3} = \frac{5 - 3}{-1} = \frac{\lambda - 2}{-4}
\]
Calculating each part:
1. For \( x \):
\[
\frac{6}{-3} = -2
\]
2. For \( y \):
\[
\frac{2}{-1} = -2
\]
3. For \( z \):
\[
\frac{\lambda - 2}{-4} = -2
\]
### Step 4: Solve for \( \lambda \)
From the equation for \( z \):
\[
\frac{\lambda - 2}{-4} = -2
\]
Cross-multiplying gives:
\[
\lambda - 2 = 8
\]
Thus,
\[
\lambda = 8 + 2 = 10
\]
### Final Answer
The value of \( \lambda \) is \( 10 \).
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