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: Given the parametric equations for lines \( L_1 \) and \( L_2 \). Line \( L_3 \) intersects \( L_1 \) and \( L_2 \) at a common point and is parallel to the vector sum \( \overrightarrow{a} + \overrightarrow{b} \).
Step 2: Determine the point of intersection of \( L_1 \) and \( L_2 \) by solving their parametric equations simultaneously. This yields the coordinates of the intersection point.
Step 3: Utilize the direction vector \( \overrightarrow{a} + \overrightarrow{b} \) and the intersection point to identify the specific coordinates that satisfy the parallelism condition for \( L_3 \). Consequently, option (A) is the correct selection.