Question

Find the shortest distance between the two parallel lines given by the vector equations: \( \vec{r} = (\hat{i} + 2\hat{j} - 4\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k}) \) and \( \vec{r} = (3\hat{i} + 3\hat{j} - 5\hat{k}) + \mu(2\hat{i} + 3\hat{j} + 6\hat{k}) \)

• A. \( \frac{\sqrt{293}}{7} \)
• B. \( \frac{\sqrt{293}}{49} \)
• C. \( 2 \)
• D. \( 0 \)

Hint:

Always check if the direction vectors are identical or proportional first. If they are proportional, use the parallel line distance formula instead of the skew lines formula to save time and avoid calculation errors.

Answer 1

The Correct Option is A

Concept: The shortest distance \( d \) between two parallel lines sharing the same direction vector \( \vec{b} \) is calculated using the cross-product formula: \[ d = \frac{|(\vec{a}_2 - \vec{a}_1) \times \vec{b}|}{|\vec{b}|} \]

Step 1: Extract vectors and find the difference vector \( \vec{a}_2 - \vec{a}_1 \). From the line equations: \[ \vec{a}_1 = \hat{i} + 2\hat{j} - 4\hat{k}, \quad \vec{a}_2 = 3\hat{i} + 3\hat{j} - 5\hat{k}, \quad \vec{b} = 2\hat{i} + 3\hat{j} + 6\hat{k} \] Subtract the position vectors: \[ \vec{a}_2 - \vec{a}_1 = (3-1)\hat{i} + (3-2)\hat{j} + (-5 - (-4))\hat{k} = 2\hat{i} + \hat{j} - \hat{k} \]

Step 2: Compute the cross product \( (\vec{a}_2 - \vec{a}_1) \times \vec{b} \). Using the matrix determinant method for cross products: \[ (\vec{a}_2 - \vec{a}_1) \times \vec{b} = \left| \begin{matrix} \hat{i} & \hat{j} & \hat{k} 2 & 1 & -1 2 & 3 & 6 \end{matrix} \right| \] \[ = \hat{i}(6 - (-3)) - \hat{j}(12 - (-2)) + \hat{k}(6 - 2) = 9\hat{i} - 14\hat{j} + 4\hat{k} \] Calculate the magnitude of this cross product: \[ |(\vec{a}_2 - \vec{a}_1) \times \vec{b}| = \sqrt{9^2 + (-14)^2 + 4^2} = \sqrt{81 + 196 + 16} = \sqrt{293} \]

Step 3: Divide by the magnitude of \( \vec{b} \) to find the distance. Find the magnitude of direction vector \( \vec{b} \): \[ |\vec{b}| = \sqrt{2^2 + 3^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \] Thus, the shortest distance is: \[ d = \frac{\sqrt{293}}{7} \]