Let \( O \) be the origin, and \( M \) and \( N \) be the points on the lines \[\frac{x - 5}{4} = \frac{y - 4}{1} = \frac{z - 5}{3} \]and\[\frac{x + 8}{12} = \frac{y + 2}{5} = \frac{z + 11}{9} \]respectively, such that \( MN \) is the shortest distance between the given lines. Then \( OM \cdot ON \) is equal to ______.
Correct Answer: 9
Solution and Explanation
Step 1: Parametrize the lines and calculate the position vectors of points \( M \) and \( N \).
The parametric equations for the two lines are: - For \( L_1 \), the point \( M \) is given by: \[ M(4\lambda + 5, \lambda + 4, 3\lambda + 5) \] - For \( L_2 \), the point \( N \) is given by: \[ N(12\mu - 8, 5\mu - 2, 9\mu - 11) \]
Step 2: Find the vector \( \overrightarrow{MN} \).
The vector \( \overrightarrow{MN} \) is given by: \[ \overrightarrow{MN} = \left( 4\lambda - 12\mu + 13, \lambda - 5\mu + 6, 3\lambda - 9\mu + 16 \right) \]
Step 3: Calculate the cross product \( \overrightarrow{b_1} \times \overrightarrow{b_2} \) of the direction vectors of the lines.
The direction vector \( \overrightarrow{b_1} \) for line \( L_1 \) is \( (4, 1, 3) \), and for line \( L_2 \), the direction vector \( \overrightarrow{b_2} \) is \( (12, 5, 9) \). The cross product is calculated as: \[ \overrightarrow{b_1} \times \overrightarrow{b_2} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 4 & 1 & 3 \\ 12 & 5 & 9 \end{vmatrix} = -6\hat{i} + 8\hat{k} \] Thus, the cross product is \( \overrightarrow{b_1} \times \overrightarrow{b_2} = (-6, 0, 8) \).
Step 4: Set up the system of equations.
Using the relation between the cross product and the shortest distance, we set up the following system: \[ \frac{4\lambda - 12\mu + 13}{-6} = \frac{\lambda - 5\mu + 6}{0} = \frac{3\lambda - 9\mu + 16}{8} \] From this, we get the two equations: \[ \lambda - 5\mu + 6 = 0 \quad \dots \text{(1)} \] \[ \lambda - 3\mu + 4 = 0 \quad \dots \text{(2)} \]
Step 5: Solve the system of equations.
Solving equations (1) and (2), we get: \[ \lambda = -1, \quad \mu = 1 \]
Step 6: Find the points \( M \) and \( N \) and compute \( OM \cdot ON \).
Substitute \( \lambda = -1 \) and \( \mu = 1 \) into the parametric equations to find the coordinates of points \( M \) and \( N \): - \( M(1, 3, 2) \) - \( N(4, 3, -2) \) The vectors \( OM \) and \( ON \) are: \[ OM = (1, 3, 2), \quad ON = (4, 3, -2) \] Now, calculate the dot product: \[ OM \cdot ON = 1 \times 4 + 3 \times 3 + 2 \times (-2) = 4 + 9 - 4 = 9 \]
Final Answer:
The value of \( OM \cdot ON \) is: \[ \boxed{9} \]
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