Key Concepts and Formulas

• Vector Equation of a Line: A line passing through a point with position vector $\vec{a}$ and parallel to a direction vector $\vec{b}$ can be represented as $\vec{r} = \vec{a} + t\vec{b}$, where $t$ is a scalar parameter.
• Intersection of Two Lines in 3D: Two lines $\vec{r} = \vec{a}_1 + l\vec{b}_1$ and $\vec{r} = \vec{a}_2 + m\vec{b}_2$ intersect if and only if there exist scalar values of $l$ and $m$ such that $\vec{a}_1 + l\vec{b}_1 = \vec{a}_2 + m\vec{b}_2$. This equality leads to a system of three linear equations in two variables ($l$ and $m$). The lines intersect if and only if this system is consistent.
• Consistency of Linear Equations: A system of linear equations is consistent if there is at least one solution. For a system of three equations in two variables, consistency means that the solution obtained from any two equations also satisfies the third equation.

Step-by-Step Solution

Step 1: Express the given lines in component form. We are given the vector equations of two lines: Line 1 ($L_1$): $\overrightarrow r = \left( {\widehat i - \widehat j} \right) + l\left( {2\widehat i + \widehat k} \right)$ Line 2 ($L_2$): $\overrightarrow r = \left( {2\widehat i - \widehat j} \right) + m\left( {\widehat i + \widehat j + \widehat k} \right)$ To find if they intersect, we need to equate their position vectors. First, let's write the position vector of a general point on each line by grouping the $\widehat i, \widehat j, \widehat k$ components.

For Line 1: $\overrightarrow r_1 = (\widehat i - \widehat j) + l(2\widehat i + \widehat k)$ $\overrightarrow r_1 = \widehat i - \widehat j + 2l\widehat i + l\widehat k$ $\overrightarrow r_1 = (1 + 2l)\widehat i + (-1)\widehat j + (l)\widehat k$

For Line 2: $\overrightarrow r_2 = (2\widehat i - \widehat j) + m(\widehat i + \widehat j + \widehat k)$ $\overrightarrow r_2 = 2\widehat i - \widehat j + m\widehat i + m\widehat j + m\widehat k$ $\overrightarrow r_2 = (2 + m)\widehat i + (-1 + m)\widehat j + (m)\widehat k$

Step 2: Set the position vectors equal to each other to represent a common intersection point. If the lines intersect, there must be a point that lies on both lines. This means there exist values of $l$ and $m$ such that $\overrightarrow r_1 = \overrightarrow r_2$. $(1 + 2l)\widehat i - \widehat j + l\widehat k = (2 + m)\widehat i + (m - 1)\widehat j + m\widehat k$

Step 3: Equate the coefficients of $\widehat i, \widehat j, \widehat k$ to form a system of linear equations. For the equality of two vectors, their corresponding components must be equal. This gives us three equations:

1. Equating $\widehat i$ coefficients: $1 + 2l = 2 + m$ $2l - m = 2 - 1$ $2l - m = 1 \quad \text{(Equation 1)}$

2. Equating $\widehat j$ coefficients: $-1 = m - 1$ $m = -1 + 1$ $m = 0 \quad \text{(Equation 2)}$

3. Equating $\widehat k$ coefficients: $l = m \quad \text{(Equation 3)}$

Step 4: Solve the system of equations for $l$ and $m$ and check for consistency. We have a system of three equations with two unknowns, $l$ and $m$. We can use two of the equations to find the values of $l$ and $m$ and then check if these values satisfy the third equation.

From Equation 2, we directly get: $m = 0$

Now, substitute $m = 0$ into Equation 3: $l = m$ $l = 0$

So, we have found potential values $l = 0$ and $m = 0$. These values were derived from Equations 2 and 3. For the lines to intersect, these values must also satisfy Equation 1.

Let's substitute $l = 0$ and $m = 0$ into Equation 1: $2l - m = 1$ $2(0) - (0) = 1$ $0 - 0 = 1$ $0 = 1$

This result, $0 = 1$, is a contradiction. This means that the values of $l$ and $m$ that satisfy two of the equations do not satisfy the third equation. Therefore, there are no values of $l$ and $m$ for which the position vectors are equal, implying that the lines do not share a common point.

Step 5: Conclude based on the consistency of the system. Since the system of equations derived from the intersection condition is inconsistent, the two lines do not intersect. They are not parallel because their direction vectors ($2\widehat i + \widehat k$ and $\widehat i + \widehat j + \widehat k$) are not proportional. Therefore, they are skew lines.

The lines do not intersect for any values of $l$ and $m$.

Common Mistakes & Tips

• Incomplete Check: A common mistake is to solve two equations and assume intersection without verifying with the third equation. In 3D, lines can appear to intersect in a 2D projection but are actually skew.
• Parallel Lines: Always check if the direction vectors are proportional first. If they are, the lines are parallel and either coincide (infinite intersection points) or are distinct (no intersection points). If they are not proportional, they can either intersect at a single point or be skew.
• Algebraic Errors: Be very careful with algebraic manipulations when solving the system of equations, as a small error can lead to an incorrect conclusion about consistency.

Summary

To determine if two lines in 3D space intersect, we set their vector equations equal to each other. This leads to a system of three linear equations for the two parameters. If this system is consistent (i.e., there are values of the parameters that satisfy all three equations), the lines intersect. If the system is inconsistent, the lines do not intersect. In this case, solving two of the equations yielded $l=0$ and $m=0$, but these values did not satisfy the third equation, indicating that the lines do not intersect.

The final answer is $\boxed{A}$.