Show that lines $\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(3\hat{i} - \hat{j})$ and $\vec{r} = (4\hat{i} - \hat{k}) + \mu(2\hat{i} + 3\hat{k})$ intersect and find point of intersection.
Step-by-Step Solution
Given: Show that lines $\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(3\hat{i} - \hat{j})$ and $\vec{r} = (4\hat{i} - \hat{k}) + \mu(2\hat{i} + 3\hat{k})$ intersect and find point of intersection.
Step 1: Formulate initial mathematical setup / substitution:
Given equation / conditions:
$$Solve \lambda and \mu$$
[1.0 Mark]
Step 2: Perform intermediate algebraic / calculus operations:
Executing the step-by-step reduction:
$$> Point of intersection (4, 0, -1).$$
[1.0 Mark]
Step 3: Evaluate final values / boundary conditions:
Substituting limits or solving the simplified system:
$$> Point of intersection (4, 0, -1).$$
[1.0 Mark]
Conclusion: The final evaluated answer is > Point of intersection (4, 0, -1)..
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🎯 Official CBSE Marking Scheme:
Formulating mathematical relation: 1.0 Mark
Executing algebraic/calculus operations: 1.0 Mark
Evaluating final solution/vector: 1.0 Mark
Correct Answer: