Show that the line passing through the points A $(0, -1, -1)$ and B $(4, 5, 1)$ intersects the line joining points C $(3, 9, 4)$ and D $(-4, 4, 4)$.
Show that the line passing through the points A $(0, -1, -1)$ and B $(4, 5, 1)$ intersects the line joining points C $(3, 9, 4)$ and D $(-4, 4, 4)$.
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Solution and Explanation
Step 1: Parametric Equations of the Lines
Line \( AB \):
Points: \( A(0, -1, -1) \), \( B(4, 5, 1) \).
Direction vector of \( AB \):
\[ \vec{AB} = (4 - 0, 5 - (-1), 1 - (-1)) = (4, 6, 2). \]
Parametric equations for line \( AB \), using point \( A \) and parameter \( s \):
\[ x = 4s, \quad y = -1 + 6s, \quad z = -1 + 2s. \]
Line \( CD \):
Points: \( C(3, 9, 4) \), \( D(-4, 4, 4) \).
Direction vector of \( CD \):
\[ \vec{CD} = (-4 - 3, 4 - 9, 4 - 4) = (-7, -5, 0). \]
Parametric equations for line \( CD \), using point \( C \) and parameter \( t \):
\[ x = 3 - 7t, \quad y = 9 - 5t, \quad z = 4. \]
Step 2: Set Up Equations for Intersection
Equate the parametric equations to find a common point:
- x-coordinate: \( 4s = 3 - 7t \)
- y-coordinate: \( -1 + 6s = 9 - 5t \)
- z-coordinate: \( -1 + 2s = 4 \)
Step 3: Solve the Equations
From the z-coordinate:
\[ -1 + 2s = 4 \implies 2s = 5 \implies s = \frac{5}{2}. \]
Substitute \( s = \frac{5}{2} \) into the x-coordinate:
\[ 4 \cdot \frac{5}{2} = 10 \implies 3 - 7t = 10 \implies -7t = 7 \implies t = -1. \]
Verify with the y-coordinate:
Left side: \( -1 + 6 \cdot \frac{5}{2} = -1 + 15 = 14 \).
Right side: \( 9 - 5 \cdot (-1) = 9 + 5 = 14 \).
The equation holds, so \( s = \frac{5}{2} \), \( t = -1 \) is consistent.
Step 4: Find the Intersection Point
For line \( AB \), using \( s = \frac{5}{2} \):
\[ x = 4 \cdot \frac{5}{2} = 10, \quad y = -1 + 6 \cdot \frac{5}{2} = 14, \quad z = -1 + 2 \cdot \frac{5}{2} = 4. \]
Point: \( (10, 14, 4) \).
For line \( CD \), using \( t = -1 \):
\[ x = 3 - 7 \cdot (-1) = 10, \quad y = 9 - 5 \cdot (-1) = 14, \quad z = 4. \]
Point: \( (10, 14, 4) \).
The points match, confirming intersection at \( (10, 14, 4) \).
Step 5: Check for Skew or Coplanar Lines
Direction vectors: \( AB: (4, 6, 2) \), \( CD: (-7, -5, 0) \).
Check if parallel by comparing ratios:
\[ \frac{4}{-7} \neq \frac{6}{-5} \neq \frac{2}{0}. \]
The lines are not parallel and share a point, so they intersect.
Final Answer
The lines intersect at:
\[ \boxed{(10, 14, 4)} \]
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