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Explain solution RD Sharma class 12 chapter Straight Line in Space exercise 27.5 question 7sub question (iv) maths

Answers (1)

Answer:  d=9

Hint: using the expression  d=\frac{\left|\left(\overrightarrow{a_{2}}-\overrightarrow{a_{1}}\right)\left(\overrightarrow{b_{1}} \times \overrightarrow{b_{2}}\right)\right|}{\left|\left(\overrightarrow{b_{1}} \times \overrightarrow{b_{2}}\right)\right|}

Given:

 

\begin{aligned} &\overrightarrow{8}=6 \hat{\imath}+2 \hat{\jmath}+2 \hat{k}+\lambda(\hat{\imath}-2 \hat{\jmath}+2 \hat{k}) \text { and } \\\\ &\overrightarrow{8}=4 \hat{\imath}-\hat{k}+\boldsymbol{\mu}(2 \hat{\imath}+3 \hat{\jmath}+\hat{k}) \end{aligned}

 

Solution:

\begin{aligned} &\overrightarrow{a_{1}}=6 \hat{\imath}+2 \hat{\jmath}+2 \hat{k} \\\\ &\overrightarrow{a_{2}}=4 \hat{\imath}-\hat{k} \\\\ &\overrightarrow{b_{1}}=\hat{\imath}-2 \hat{\jmath}+2 \hat{k} \end{aligned}

\begin{aligned} &\overrightarrow{b_{2}}=\hat{3}_{l}-2 \hat{\jmath}-2 \hat{k} \\\\ &\overrightarrow{a_{2}}-\overrightarrow{a_{1}}=-10 \hat{\imath}-2 \hat{\jmath}+3 \hat{k} \end{aligned}

\text { and } \overrightarrow{b_{1}} \times \overrightarrow{b_{2}}=\left|\begin{array}{ccc} \hat{\imath} & \hat{\jmath} & \hat{k} \\ 1 & -2 & 2 \\ 3 & -2 & -2 \end{array}\right| \Rightarrow 8 \hat{\imath}+8 \hat{\jmath}+4 \hat{k}

\begin{aligned} &\Rightarrow\left|\overrightarrow{b_{1}} \times \overline{b_{2}}\right| \Rightarrow \sqrt{8^{2}+8^{2}+4^{2}} \Rightarrow \sqrt{64+64+16} \\\\ &=\sqrt{144} \Rightarrow 12 \end{aligned}

and

\begin{aligned} &\left(\overrightarrow{a_{2}}-\overrightarrow{a_{1}}\right)\left(\overrightarrow{b_{1}} \times \overrightarrow{b_{2}}\right)=-10 \hat{\imath}-2 \hat{\jmath}+3 \hat{k}(8 \hat{\imath}+8 \hat{\jmath}+4 \hat{k}) \\\\ &=-80-16-12 \\\\ &=-108 \end{aligned}

The shortest distance between the lines

\begin{aligned} &\overrightarrow{8}=\overrightarrow{a_{1}}+\lambda \overrightarrow{b_{1}} \text { and } \overrightarrow{8}=\overrightarrow{a_{2}}+\mu b_{2} \\\\ &\mathrm{~d}=\left|\frac{-108}{12}\right| \\\\ &=9 \end{aligned}

 

 

 

 

 

 

Posted by

infoexpert26

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