# Find the distance between the lines l1 and l2 with the following vector equations. $$\vec{r}=2\hat{i}+2\hat{j}-2\hat{k}+λ(3\hat{i}+2\hat{j}+5\hat{k})$$ $$\vec{r}=4 \hat{i}-\hat{j}+5\hat{k}+μ(3\hat{i}-2\hat{j}+4\hat{k})$$

Category: QuestionsFind the distance between the lines l1 and l2 with the following vector equations. $$\vec{r}=2\hat{i}+2\hat{j}-2\hat{k}+λ(3\hat{i}+2\hat{j}+5\hat{k})$$ $$\vec{r}=4 \hat{i}-\hat{j}+5\hat{k}+μ(3\hat{i}-2\hat{j}+4\hat{k})$$
Editor">Editor Staff asked 11 months ago

Find the distance between the lines l1 and l2 with the following vector equations.

\vec{r}=2\hat{i}+2\hat{j}-2\hat{k}+λ(3\hat{i}+2\hat{j}+5\hat{k})

\vec{r}=4 \hat{i}-\hat{j}+5\hat{k}+μ(3\hat{i}-2\hat{j}+4\hat{k})

(a) \frac{57}{\sqrt{47}}

(b) \frac{57}{\sqrt{77}}

(c) \frac{7}{\sqrt{477}}

(d) \frac{57}{\sqrt{477}}

I got this question in semester exam.

The query is from Three Dimensional Geometry in section Three Dimensional Geometry of Mathematics – Class 12
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options
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Editor">Editor Staff answered 11 months ago

Right option is (d) \frac{57}{\sqrt{477}}

To explain I would say: We know that, the shortest distance between two skew lines is given by

d=\left |\frac{(\vec{b_1}×\vec{b_2}).(a_2-a_1)}{|\vec{b_1}×\vec{b_2}|}\right |

The vector equations of the two lines is

\vec{r}=2\hat{i}+2\hat{j}-2\hat{k}+λ(3\hat{i}+2\hat{j}+5\hat{k})

\vec{r}=4 \hat{i}-\hat{j}+5\hat{k}+μ(3\hat{i}-2\hat{j}+4\hat{k})

∴d=\left|\frac{((3\hat{i}+2\hat{j}+5\hat{k})×(3\hat{i}-2\hat{j}+4\hat{k}).(4\hat{i}-\hat{j}+5\hat{k})-(2\hat{i}+2\hat{j}-2\hat{k}))}{|(3\hat{i}+2\hat{j}+5\hat{k})×(3\hat{i}-2\hat{j}+4\hat{k})|}\right |

(3\hat{i}+2\hat{j}+5\hat{k})×(3\hat{i}-2\hat{j}+4\hat{k})=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\3&2&5\\3&-2&4\end{vmatrix}

=\hat{i}(8+10)-\hat{j}(12-15)+\hat{k}(-6-6)

=18\hat{i}+3\hat{j}-12\hat{k}

d=\left|\frac{(18\hat{i}+3\hat{j}-12\hat{k}).(2\hat{i}-3\hat{j}+7\hat{k})}{\sqrt{18^2+3^2+(-12)^2}}\right |

d=\left|\frac{36-9-84}{\sqrt{477}}\right |=\frac{57}{\sqrt{477}}.