Let $\vec{a} = \hat{i} + 2\hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} + 7\hat{j} + 3\hat{k}$.
Let
$$ L_1 : \vec{r} = (-\hat{i} + 2\hat{j} + \hat{k}) + \lambda \vec{a}, \quad \lambda \in \mathbb{R} $$
and
$$ L_2 : \vec{r} = (\hat{j} + \hat{k}) + \mu \vec{b}, \quad \mu \in \mathbb{R} $$
be two lines. If the line $L_3$ passes through the point of intersection of $L_1$ and $L_2$,
and is parallel to $\vec{a} + \vec{b}$, then $L_3$ passes through the point:
Let $$ L_1 : \vec{r} = (-\hat{i} + 2\hat{j} + \hat{k}) + \lambda \vec{a}, \quad \lambda \in \mathbb{R} $$ and $$ L_2 : \vec{r} = (\hat{j} + \hat{k}) + \mu \vec{b}, \quad \mu \in \mathbb{R} $$ be two lines. If the line $L_3$ passes through the point of intersection of $L_1$ and $L_2$, and is parallel to $\vec{a} + \vec{b}$, then $L_3$ passes through the point:
Show Hint
- \( (-1, -1, 1) \)
- \( (5, 17, 4) \)
- \( (2, 8, 5) \)
- \( (8, 26, 12) \)
The Correct Option is A
Solution and Explanation
Step 1: The parametric equations of the lines \( L_1 \) and \( L_2 \) are given, and the line \( L_3 \) passes through their point of intersection and is parallel to \( \overrightarrow{a} + \overrightarrow{b} \).
Step 2: To find the point of intersection, we need to solve the system of equations given by the parametric equations of \( L_1 \) and \( L_2 \). After solving this, we obtain the coordinates of the intersection point.
Step 3: Since \( L_3 \) is parallel to \( \overrightarrow{a} + \overrightarrow{b} \), we use this direction vector and the intersection point to identify the correct coordinates that satisfy the condition. Thus, the correct answer is (A).
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