Question

If \( \bar{a} = \bar{i} - \bar{j} + 3\bar{k} \), \( \bar{c} = -\bar{k} \) are position vectors of two points and \( \bar{b} = 2\bar{i} - \bar{j} + \lambda \bar{k} \),

\( \bar{d} = \bar{i} + \bar{j} - \bar{k} \) are two vectors, then the lines \( r = \bar{a} + t \bar{b} \), \( r = \bar{c} + s \bar{d} \) are:

• (1) Skew lines when \( \lambda \neq \frac{19}{3} \)
• (2) Coplanar \( \forall \lambda \in \mathbb{R} \)
• (3) Skew lines when \( \lambda \neq \frac{19}{3} \)
• (4) Coplanar when \( \lambda = \frac{19}{3} \)

Hint:

To check whether two lines are coplanar, compute the determinant of the matrix formed by their direction vectors and the displacement vector between them. If the determinant is zero, the lines are coplanar.

Answer 1

The Correct Option is C

Step 1: Determine the Direction Vectors of the Given Lines

The direction vectors of the given lines are:

\[ \bar{b} = 2\bar{i} - \bar{j} + \lambda \bar{k}, \quad \bar{d} = \bar{i} + \bar{j} - \bar{k}. \]

Step 2: Condition for Coplanarity

Two lines are coplanar if their direction vectors and the vector joining a point on one line to a point on the other line are linearly dependent. The vector joining points \( \bar{a} \) and \( \bar{c} \):

\[ \bar{AC} = \bar{c} - \bar{a} = (-\bar{k}) - (\bar{i} - \bar{j} + 3\bar{k}) = -\bar{i} + \bar{j} - 4\bar{k}. \]

The three vectors \( \bar{b} \), \( \bar{d} \), and \( \bar{AC} \) must be linearly dependent for the lines to be coplanar. This requires:

\[ \begin{vmatrix} 2 & -1 & \lambda \\ 1 & 1 & -1 \\ -1 & 1 & -4 \end{vmatrix} = 0. \]

Step 3: Solve for \( \lambda \)

Expanding the determinant:

\[ 2 \begin{vmatrix} 1 & -1 \\ 1 & -4 \end{vmatrix} - (-1) \begin{vmatrix} 1 & -1 \\ -1 & -4 \end{vmatrix} + \lambda \begin{vmatrix} 1 & 1 \\ -1 & 1 \end{vmatrix} = 0. \] \[ 2[(1)(-4) - (-1)(1)] + 1[(1)(-4) - (-1)(-1)] + \lambda [(1)(1) - (1)(-1)] = 0. \] \[ 2(-4 + 1) + (-4 - 1) + \lambda(1 + 1) = 0. \] \[ 2(-3) + (-5) + 2\lambda = 0. \] \[ -6 - 5 + 2\lambda = 0. \] \[ 2\lambda = 11. \] \[ \lambda = \frac{19}{3}. \]

Step 4: Conclusion

• If \( \lambda = \frac{19}{3} \), the lines are coplanar.
• If \( \lambda \neq \frac{19}{3} \), the lines are skew.