Name: The Shortest Distance Between Two Skew Lines
Uploaded: 2019-07-19
Duration: 9 min 21 s
Description: Skew lines are the lines that do not intersect and not parallel. The shortest distance between two skew lines lies along the line, which is perpendicular to...

Question

Find the shortest distance between the following pairs of parallel lines whose equations are:

\vec{r} = (\hat{i} + 2 \hat{j} + 3 \hat{k}) + λ (\hat{i} - \hat{j} + \hat{k})

and

\vec{r} = (2 \hat{i} - \hat{j} - \hat{k}) + μ (- \hat{i} + \hat{j} - \hat{k})

Accepted Answer

The vector equations of the given lines are

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

$\vec{r} = (2 \hat{i} - \hat{j} - \hat{k}) + μ (- \hat{i} + \hat{j} - \hat{k})$

$= (2 \hat{i} - \hat{j} - \hat{k}) - μ (\hat{i} - \hat{j} + \hat{k}) \dots (2)$

These two lines pass through the points having position vectors $\vec{r} = (\hat{i} + 2 \hat{j} + 3 \hat{k}) + λ (\hat{i} - \hat{j} + \hat{k})$ and

$\vec{a_{2}} = 2 \hat{i} - \hat{j} - \hat{k}$ and are parallel to the vector $\vec{b} = \hat{i} - \hat{j} + \hat{k}$

Now, $\vec{a_{2}} - \vec{a_{1}} = \hat{i} - 3 \hat{j} - 4 \hat{k}$ and

$(\vec{a_{2}} - \vec{a_{1}}) \times \vec{b} = (\hat{i} - 3 \hat{j} - 4 \hat{k}) \times (\hat{i} - \hat{j} + \hat{k})$

$= |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & - 3 & - 4 \\ 1 & - 1 & 1 \end{matrix}|$

$= \hat{i} ((- 3 \times 1) - (- 4 \times - 1)) - \hat{j} ((1 \times 1) - (- 4 \times 1)) + \hat{k} ((1 \times - 1) - (- 3 \times 1))$

$= - 7 \hat{i} - 5 \hat{j} + 2 \hat{k}$

$\Rightarrow |(\vec{a_{2}} - \vec{a_{1}}) \times \vec{b}| = \sqrt{(- 7)^{2} + (- 5)^{2} + 2^{2}}$

$= \sqrt{49 + 25 + 4}$

$= \sqrt{78}$

The shortest distance between the two lines is given by

$\frac{|(\vec{a_{2}} - \vec{a_{1}}) \times \vec{b}|}{| \vec{b} |} = \frac{\sqrt{78}}{\sqrt{3}} = \sqrt{26}$

Find the shortest distance between the following pairs of parallel lines whose equations are: $\vec{r} = (\hat{i} + 2 \hat{j} + 3 \hat{k}) + λ (\hat{i} - \hat{j} + \hat{k})$ and $\vec{r} = (2 \hat{i} - \hat{j} - \hat{k}) + μ (- \hat{i} + \hat{j} - \hat{k})$

Important Questions on Straight Line in Space

Learn and Prepare for any exam you want !

Find the shortest distance between the following pairs of parallel lines whose equations are: r→=(i^+2j^+3k^)+λ(i^-j^+k^) and r→=(2i^-j^-k^)+μ(-i^+j^-k^)

Important Questions on Straight Line in Space

Learn and Prepare for any exam you want !

Find the shortest distance between the following pairs of parallel lines whose equations are: $\vec{r} = (\hat{i} + 2 \hat{j} + 3 \hat{k}) + λ (\hat{i} - \hat{j} + \hat{k})$ and $\vec{r} = (2 \hat{i} - \hat{j} - \hat{k}) + μ (- \hat{i} + \hat{j} - \hat{k})$