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  • Find the shortest distance between the lines r vector = i caret +2 j caret + k caret + lambda i caret - j caret + k caret and r vector =2 i caret - j caret -k caret + mu 2 i caret+j caret + 2 k caret

14     Find the shortest distance between the lines

              \overrightarrow{r}=(\widehat{i}+2\widehat{j}+\widehat{k})+\lambda (\widehat{i}-\widehat{j}+\widehat{k}) and  \overrightarrow{r}=(2\widehat{i}-\widehat{j}-\widehat{k})+\mu (2\widehat{i}+\widehat{j}+2\widehat{k})

                  

Answers (1)

best_answer

So given equation of lines;

 \overrightarrow{r}=(\widehat{i}+2\widehat{j}+\widehat{k})+\lambda (\widehat{i}-\widehat{j}+\widehat{k}) and  \overrightarrow{r}=(2\widehat{i}-\widehat{j}-\widehat{k})+\mu (2\widehat{i}+\widehat{j}+2\widehat{k}) in the vector form.

Now, we can find the shortest distance between the lines \vec{r} = \vec{a_{1}}+\lambda\vec{b_{1}} and \vec{r} = \vec{a_{2}}+\mu \vec{b_{2}}, is given by the formula,

d= \left | \frac{\left ( \vec{b_{1}}\times\vec{b_{2}} \right ).\left ( \vec{a_{2}}-\vec{a_{1}} \right )}{\left | \vec{b_{1}}\times \vec{b_{2}} \right |} \right |

Now comparing the values from the equation, we obtain

\vec{a_{1}} = \widehat{i}+2\widehat{j}+\widehat{k}                 \vec{b_{1}} = \widehat{i}-\widehat{j}+\widehat{k}

\vec{a_{2}} = 2\widehat{i}-\widehat{j}-\widehat{k}               \vec{b_{2}} = 2\widehat{i}+\widehat{j}+2\widehat{k}

\vec{a_{2}} -\vec{a_{1}} =\left ( 2\widehat{i}-\widehat{j}-\widehat{k} \right ) - \left ( \widehat{i}+2\widehat{j}+\widehat{k} \right ) = \widehat{i}-3\widehat{j}-2\widehat{k}

Then calculating 

\vec{b_{1}}\times \vec{b_{2}} = \begin{vmatrix} \widehat{i} &\widehat{j} &\widehat{k} \\ 1 & -1 &1 \\ 2& 1 &2 \end{vmatrix}

\vec{b_{1}}\times \vec{b_{2}} = (-2-1)\widehat{i} - (2-2) \widehat{j} +(1+2) \widehat{k} = -3\widehat{i}+3\widehat{k}

\Rightarrow \left | \vec{b_{1}}\times \vec{b_{2}} \right | = \sqrt{(-3)^2+(3)^2} = \sqrt{9+9} =\sqrt{18} =3\sqrt2

So, substituting the values now in the formula above we get;

d =\left | \frac{\left ( -3\widehat{i}+3\widehat{k} \right ).(\widehat{i}-3\widehat{j}-2\widehat{k})}{3\sqrt2} \right |

\Rightarrow d = \left | \frac{-3.1+3(-2)}{3\sqrt2} \right |

d = \left | \frac{-9}{3\sqrt2} \right | = \frac{3}{\sqrt2} = \frac{3\sqrt2}{2}

Therefore, the shortest distance between the two lines is \frac{3\sqrt2}{2} units.

 

 

 

Posted by

Divya Prakash Singh

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