Question

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:

• A. \( (-1, -1, 1) \)
• B. \( (5, 17, 4) \)
• C. \( (2, 8, 5) \)
• D. \( (8, 26, 12) \)

Hint:

To find the point of intersection of two lines, equate their parametric equations and solve the system of equations. The direction of the line passing through the intersection point can be determined by the sum of the direction vectors of the two lines.

Answer 1

The Correct Option is A

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).