The problem asks to find the equation of a line $g$ that intersects two given lines, $h$ and $k$, and is also parallel to line $j$.

Here’s a breakdown of the given information:

1. Line $h$ is defined by the equation: $2x + y - 1 = 0$

2. Line $k$ is defined by the equation: $3x - y + z + 2 = 0$

3. Line $j$ has the parametric form: $\frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ This suggests that the direction ratios of line $j$ are proportional to $1:2:3$.

Steps to solve:

1. Find direction ratios of line $j$ from its parametric form, which are $(1, 2, 3)$.

2. Determine the vector equation of the required line $g$, which must be parallel to line $j$. Since line $g$ is parallel to $j$, the direction ratios of $g$ should also be $(1, 2, 3)$.

3. Find intersection points of line $g$ with lines $h$ and $k$. To do this, express the parametric equations for line $g$ using the direction ratios and plug them into the equations for $h$ and $k$. Solving this system will give the required points.

Would you like a more detailed step-by-step calculation for solving this system, or do you have a specific question regarding the approach?

Additional questions:

1. How do you extract the parametric form of a line from a system of equations?
2. What is the geometric interpretation of two lines being parallel?
3. Can you explain how direction ratios help in defining the slope of a line in 3D?
4. How would you solve the system of equations for intersection points in 3D geometry?
5. Why do we equate the direction ratios when lines are parallel?

Tip:

When dealing with lines in 3D, always try to first extract direction ratios to simplify your approach toward finding intersection points or determining parallelism.