Question

The vector equations of two lines are given as:

Line 1: \[ \vec{r}_1 = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(4\hat{i} + 6\hat{j} + 12\hat{k}) \]

Line 2: \[ \vec{r}_2 = 3\hat{i} + 3\hat{j} - 5\hat{k} + \mu(6\hat{i} + 9\hat{j} + 18\hat{k}) \]

Determine whether the lines are parallel, intersecting, skew, or coincident. If they are not coincident, find the shortest distance between them.

Answer 1

Given Vector Equations of Lines

Line 1: \[ \vec{r}_1 = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(4\hat{i} + 6\hat{j} + 12\hat{k}) \] Line 2: \[ \vec{r}_2 = 3\hat{i} + 3\hat{j} - 5\hat{k} + \mu(6\hat{i} + 9\hat{j} + 18\hat{k}) \]

Step 1: Direction Vectors

\[ \vec{d}_1 = \langle 4, 6, 12 \rangle,\quad \vec{d}_2 = \langle 6, 9, 18 \rangle \] Since \[ \vec{d}_2 = \frac{3}{2} \cdot \vec{d}_1, \] the direction vectors are scalar multiples ⇒ lines are parallel.

Step 2: Check for Coincidence

Take position vectors of a point on each line: \[ \vec{a}_1 = \langle 1, 2, -4 \rangle,\quad \vec{a}_2 = \langle 3, 3, -5 \rangle \] Compute: \[ \vec{a}_2 - \vec{a}_1 = \langle 2, 1, -1 \rangle \] Check if this is a scalar multiple of \( \vec{d}_1 \): It is not, so lines are parallel but not coincident.

Step 3: Find Shortest Distance Between Lines

Use formula: \[ \text{Distance} = \frac{| (\vec{a}_2 - \vec{a}_1) \times \vec{d}_1 |}{|\vec{d}_1|} \] Let: \[ \vec{a}_2 - \vec{a}_1 = \langle 2, 1, -1 \rangle,\quad \vec{d}_1 = \langle 4, 6, 12 \rangle \] Compute cross product: \[ \vec{v} \times \vec{d}_1 = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 1 & -1 \\ 4 & 6 & 12 \end{vmatrix} = \hat{i}(1 \cdot 12 - (-1) \cdot 6) - \hat{j}(2 \cdot 12 - (-1) \cdot 4) + \hat{k}(2 \cdot 6 - 1 \cdot 4) \] \[ = \hat{i}(12 + 6) - \hat{j}(24 + 4) + \hat{k}(12 - 4) = \hat{i}(18) - \hat{j}(28) + \hat{k}(8) \Rightarrow \vec{N} = \langle 18, -28, 8 \rangle \] \[ |\vec{N}| = \sqrt{18^2 + 28^2 + 8^2} = \sqrt{324 + 784 + 64} = \sqrt{1172} \] \[ |\vec{d}_1| = \sqrt{4^2 + 6^2 + 12^2} = \sqrt{16 + 36 + 144} = \sqrt{196} = 14 \] \[ \text{Distance} = \frac{\sqrt{1172}}{14} \approx \frac{34.24}{14} \approx \boxed{2.45} \]

✅ Final Conclusion:

• Lines are parallel
• They are not coincident
• Shortest distance between them is approximately: \[ \boxed{2.45} \]