Question

A line passing through the points \( A(1, 2, 3) \) and \( B(5, 8, 11) \) intersects the line \( \vec{r} = 4\hat{i} + \hat{j} + \lambda(5\hat{i} + 2\hat{j} + \hat{k}) \). Find the co-ordinates of the point of intersection. Hence, write the equation of a line passing through the point of intersection and perpendicular to both the lines.

Hint:

Use reduced direction ratios for simpler algebra during the cross product.

Answer 1

Concept:
• Equation of a line through two points \( \vec{a} \) and \( \vec{b} \) is \( \vec{r} = \vec{a} + \mu(\vec{b} - \vec{a}) \).
• Perpendicular direction is found using the cross product of direction vectors.

Step 1: Find the equation of the first line and point of intersection
Line 1: \( \vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \mu(4\hat{i} + 6\hat{j} + 8\hat{k}) \).
Equating components with Line 2:
\[ 1 + 4\mu = 4 + 5\lambda \]
\[ 2 + 6\mu = 1 + 2\lambda \]
Solving gives \( \mu = -1/2 \) and \( \lambda = -1 \). (Using reduced DRs \( 2, 3, 4 \) is easier).
Point of intersection: \( (-1, -1, -1) \).

Step 2: Find the perpendicular direction
DRs of Line 1 are \( (2, 3, 4) \), Line 2 are \( (5, 2, 1) \).
Direction \( = (2, 3, 4) \times (5, 2, 1) \):
\[ \vec{n} = \hat{i}(3-8) - \hat{j}(2-20) + \hat{k}(4-15) = -5\hat{i} + 18\hat{j} - 11\hat{k} \]

Step 3: Write the final line equation
Equation: \( \frac{x+1}{-5} = \frac{y+1}{18} = \frac{z+1}{-11} \).