Name: Coplanar Equilibrium
Uploaded: 2019-07-19
Duration: 14 min 12 s
Description: In this video, we will learn the concept of condition for coplanarity of two lines and the equation of plane containing two non-parallel coplanar lines.

Question

Find the shortest distance between the lines

r = (4 \hat{i} - \hat{j}) + λ (\hat{i} + 2 \hat{j} - 3 \hat{k})

and

r = (\hat{i} - \hat{j} + 2 \hat{k}) + μ (\hat{i} + 4 \hat{j} - 5 \hat{k})

Accepted Answer

We know that,

The shortest distance between the lines $\vec{r} = {\vec{a}}_{1} + λ {\vec{b}}_{1}$ and $\vec{r} = {\vec{a}}_{2} + λ {\vec{b}}_{2}$ is given by
$d = |\frac{({\vec{a}}_{2} - {\vec{a}}_{1}) \cdot ({\vec{b}}_{1} \times {\vec{b}}_{2})}{|{\vec{b}}_{1} \times {\vec{b}}_{2}|}|$

Here, we have, ${\vec{a}}_{1} = 4 \hat{i} - \hat{j}, {\vec{a}}_{2} = \hat{i} - \hat{j} + 2 \hat{k}, {\vec{b}}_{1} = \hat{i} + 2 \hat{j} - 3 \hat{k}$ and ${\vec{b}}_{2} = \hat{i} + 4 \hat{j} - 5 \hat{k}$

Now, ${\vec{a}}_{2} - {\vec{a}}_{1} = \hat{i} - \hat{j} + 2 \hat{k} - 4 \hat{i} + \hat{j} = - 3 \hat{i} + 0 \hat{j} + 2 \hat{k}$

And ${\vec{b}}_{1} \times {\vec{b}}_{2} = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & - 3 \\ 1 & 4 & - 5 \end{matrix}| = (- 10 + 12) \hat{i} - (- 5 + 3) \hat{j} + (4 - 2) \hat{k} = 2 \hat{i} + 2 \hat{j} + 2 \hat{k}$

Therefore, $({\vec{a}}_{2} - {\vec{a}}_{1}) \cdot ({\vec{b}}_{1} \times {\vec{b}}_{2}) = (- 3 \hat{i} + 0 \hat{j} + 2 \hat{k}) \cdot (2 \hat{i} + 2 \hat{j} + 2 \hat{k})$

$= - 6 + 0 + 4$

$= - 2$

And $|{\vec{b}}_{1} \times {\vec{b}}_{2}| = \sqrt{4 + 4 + 4} = 2 \sqrt{3}$

Therefore,
$d = |\frac{({\vec{a}}_{2} - {\vec{a}}_{1}) \cdot ({\vec{b}}_{2} \times {\vec{b}}_{1})}{|{\vec{b}}_{2} \times {\vec{b}}_{1}|}| = |\frac{- 2}{2 \sqrt{3}}| = \frac{1}{\sqrt{3}}$ units.

Hence, the shortest distance between the lines is $\frac{1}{\sqrt{3}}$ units.

Find the shortest distance between the lines $r = (4 \hat{i} - \hat{j}) + λ (\hat{i} + 2 \hat{j} - 3 \hat{k})$ and $r = (\hat{i} - \hat{j} + 2 \hat{k}) + μ (\hat{i} + 4 \hat{j} - 5 \hat{k})$

Important Questions on Line and Plane

Learn and Prepare for any exam you want !

Find the shortest distance between the lines r=4i^-j^+λi^+2j^-3k^ and r=i^-j^+2k^+μi^+4j^-5k^

Important Questions on Line and Plane

Learn and Prepare for any exam you want !

Find the shortest distance between the lines $r = (4 \hat{i} - \hat{j}) + λ (\hat{i} + 2 \hat{j} - 3 \hat{k})$ and $r = (\hat{i} - \hat{j} + 2 \hat{k}) + μ (\hat{i} + 4 \hat{j} - 5 \hat{k})$