Question

The position vector of the point of intersection of the line joining the points \( \mathbf{i} - \mathbf{j} + \mathbf{k} \) and the line joining the points \( 2\mathbf{i} + \mathbf{j} - 6\mathbf{k} \), \( 3\mathbf{i} - \mathbf{j} - 7\mathbf{k} \) is:

• A. \( \mathbf{i} - 3\mathbf{j} + 4\mathbf{k} \)
• B. \( 4\mathbf{i} - 3\mathbf{j} - 8\mathbf{k} \)
• C. \( \mathbf{i} + 3\mathbf{j} - 5\mathbf{k} \)
• D. \( \mathbf{i} + \mathbf{j} - 2\mathbf{k} \) \bigskip

Hint:

When solving for the intersection of two lines in 3D, equate the parametric equations and solve for consistent parameter values. If the parameters differ across equations, the lines are skew and do not intersect.

Answer 1

The Correct Option is C

We are given two lines: The first line passes through the points \( A(\mathbf{i} - \mathbf{j} + \mathbf{k}) \) and \( B(2\mathbf{i} + \mathbf{j} - 6\mathbf{k}) \).
The second line passes through the points \( C(3\mathbf{i} - \mathbf{j} - 7\mathbf{k}) \) and \( D(\mathbf{i} - \mathbf{j} + \mathbf{k}) \). Our objective is to determine the point where these two lines intersect. 

Step 1: Derive the parametric equations of both lines. For the line passing through \( A \) and \( B \), the parametric form is: \[ \mathbf{r}_1 = (1 - t) \mathbf{A} + t \mathbf{B} = (1 - t)(\mathbf{i} - \mathbf{j} + \mathbf{k}) + t(2\mathbf{i} + \mathbf{j} - 6\mathbf{k}). \] Expanding: \[ \mathbf{r}_1 = \mathbf{i} - \mathbf{j} + \mathbf{k} - t \mathbf{i} + t \mathbf{j} - t \mathbf{k} + 2t \mathbf{i} + t \mathbf{j} - 6t \mathbf{k}. \] \[ \mathbf{r}_1 = (1 + t) \mathbf{i} + (2t) \mathbf{j} + (1 - t) \mathbf{k}. \] For the second line passing through \( C \) and \( D \), the parametric equation is: \[ \mathbf{r}_2 = (1 - t) \mathbf{C} + t \mathbf{D} = (1 - t)(3\mathbf{i} - \mathbf{j} - 7\mathbf{k}) + t(\mathbf{i} - \mathbf{j} + \mathbf{k}). \] Expanding: \[ \mathbf{r}_2 = (3 - 3t) \mathbf{i} - (1 - 2t) \mathbf{j} + (-7 + 8t) \mathbf{k}. \] 

Step 2: Find the intersection by equating parametric equations. Setting \( \mathbf{r}_1 = \mathbf{r}_2 \), we obtain: \[ (1 + t) \mathbf{i} + (2t) \mathbf{j} + (1 - t) \mathbf{k} = (3 - 3t) \mathbf{i} - (1 - 2t) \mathbf{j} + (-7 + 8t) \mathbf{k}. \] Equating coefficients for \( \mathbf{i}, \mathbf{j}, \mathbf{k} \), we get: 1. \( 1 + t = 3 - 3t \) 2. \( 2t = -1 + 2t \) 3. \( 1 - t = -7 + 8t \) 

 Step 3: Solve the system of equations. - The equation \( 2t = -1 + 2t \) is always valid, giving no useful information. - Solving \( 1 + t = 3 - 3t \): \[ 1 + t = 3 - 3t \quad \Rightarrow \quad 4t = 2 \quad \Rightarrow \quad t = \frac{1}{2}. \] - Solving \( 1 - t = -7 + 8t \): \[ 1 - t = -7 + 8t \quad \Rightarrow \quad 9t = 8 \quad \Rightarrow \quad t = \frac{8}{9}. \] Since the values of \( t \) do not match, the lines do not intersect at a common point. Thus, we conclude that the lines are skew. 

Conclusion: Since the equations lead to inconsistent parameter values, the two lines do not intersect in three-dimensional space. \bigskip