The distance between the lines $\vec{r}=\hat{i}-2 \hat{j}+3 \hat{k}+\lambda(2 \hat{i}+3 \hat{j}+6 \hat{k})$ and $\vec{r}=3 \hat{i}-2 \hat{j}+1 \hat{k}+\mu(4 \hat{i}+6 \hat{j}+12 \hat{k})$ is :

Distance Between Two Lines in 3D Space

Given Information:

• Line 1: $\vec{r} = \hat{i} - 2\hat{j} + 3\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k})$
  • Line 2: $\vec{r} = 3\hat{i} - 2\hat{j} + 1\hat{k} + \mu(4\hat{i} + 6\hat{j} + 12\hat{k})$

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Identify the direction vectors and points on each line.

Line 1:

• Direction vector: $\vec{a} = 2\hat{i} + 3\hat{j} + 6\hat{k}$
  • Point on line: $\vec{r}_1 = \hat{i} - 2\hat{j} + 3\hat{k}$

Line 2:

• Direction vector: $\vec{b} = 4\hat{i} + 6\hat{j} + 12\hat{k}$
  • Point on line: $\vec{r}_2 = 3\hat{i} - 2\hat{j} + 1\hat{k}$

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Check if the direction vectors are parallel.

$\vec{b} = 4\hat{i} + 6\hat{j} + 12\hat{k} = 2(2\hat{i} + 3\hat{j} + 6\hat{k}) = 2\vec{a}$

Since $\vec{b} = 2\vec{a}$, the lines are parallel.

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Calculate the distance between parallel lines using the formula:

$d = \dfrac{|(\vec{r}_2 - \vec{r}_1) \times \vec{a}|}{|\vec{a}|}$

Find $\vec{r}_2 - \vec{r}_1$:

$\vec{r}_2 - \vec{r}_1 = (3\hat{i} - 2\hat{j} + 1\hat{k}) - (\hat{i} - 2\hat{j} + 3\hat{k})$

$\vec{r}_2 - \vec{r}_1 = 2\hat{i} - 2\hat{k}$

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Calculate $(\vec{r}_2 - \vec{r}_1) \times \vec{a}$:

$(\vec{r}_2 - \vec{r}_1) \times \vec{a} = (2\hat{i} - 2\hat{k}) \times (2\hat{i} + 3\hat{j} + 6\hat{k})$

Using the cross product formula:

$\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 0 & -2 \\ 2 & 3 & 6 \end{vmatrix}$

$= \hat{i}(0 \times 6 - (-2) \times 3) - \hat{j}(2 \times 6 - (-2) \times 2) + \hat{k}(2 \times 3 - 0 \times 2)$

$= \hat{i}(0 + 6) - \hat{j}(12 + 4) + \hat{k}(6 - 0)$

$= 6\hat{i} - 16\hat{j} + 6\hat{k}$

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Calculate $|(\vec{r}_2 - \vec{r}_1) \times \vec{a}|$:

$|6\hat{i} - 16\hat{j} + 6\hat{k}| = \sqrt{6^2 + (-16)^2 + 6^2} = \sqrt{36 + 256 + 36} = \sqrt{328}$

Calculate $|\vec{a}|$:

$|\vec{a}| = |2\hat{i} + 3\hat{j} + 6\hat{k}| = \sqrt{2^2 + 3^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7$

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Calculate the distance:

$d = \dfrac{|(\vec{r}_2 - \vec{r}_1) \times \vec{a}|}{|\vec{a}|} = \dfrac{\sqrt{328}}{7}$