Write the condition for the lines \[\vec{r} = \overrightarrow{a_1} + \lambda \overrightarrow{b_1} \text{ and } \overrightarrow{r} = \overrightarrow{a_2} + \mu \overrightarrow{b_2}\] to be intersecting.

The shortest distance between the lines \[\overrightarrow{r} = \overrightarrow{a_1} + \lambda \overrightarrow{b_1} \text{ and } \overrightarrow{r} = \overrightarrow{a_2} + \mu \overrightarrow{b_2}\] is given by \[d = \left| \frac{\left( \overrightarrow{a_2} - \overrightarrow{a_1} \right) . \left( \overrightarrow{b_1} \times \overrightarrow{b_2} \right)}{\left| \overrightarrow{b_1} \times \overrightarrow{b_2} \right|} \right|\] For the lines to be intersecting, d = 0 . \[\Rightarrow \frac{\left( \overrightarrow{a_2} - \overrightarrow{a_1} \right) . \left( \overrightarrow{b_1} \times \overrightarrow{b_2} \right)}{\left| \overrightarrow{b_1} \times \overrightarrow{b_2} \right|} = 0\] \[ \Rightarrow \left( \overrightarrow{a_2} - \overrightarrow{a_1} \right) . \left( \overrightarrow{b_1} \times \vec{b_2} \right) = 0\]

Write the Condition for the Lines → R = → a 1 + λ → B 1 and → R = → a 2 + μ → B 2 to Be Intersecting. - Mathematics

प्रश्न

Write the condition for the lines $\vec{r} = \vec{a_{1}} + λ \vec{b_{1}} and \vec{r} = \vec{a_{2}} + μ \vec{b_{2}}$ to be intersecting.

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उत्तर

The shortest distance between the lines

$\vec{r} = \vec{a_{1}} + λ \vec{b_{1}} and \vec{r} = \vec{a_{2}} + μ \vec{b_{2}}$ is given by

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

For the lines to be intersecting,

d = 0 .

$\Rightarrow \frac{(\vec{a_{2}} - \vec{a_{1}}) . (\vec{b_{1}} \times \vec{b_{2}})}{| \vec{b_{1}} \times \vec{b_{2}} |} = 0$

$\Rightarrow (\vec{a_{2}} - \vec{a_{1}}) . (\vec{b_{1}} \times \vec{b_{2}}) = 0$

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Equation of a Line in Space

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Q 14 Q 13 Q 15

अध्याय 28: Straight Line in Space - Very Short Answers [पृष्ठ ४१]

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अध्याय 28 Straight Line in Space
Very Short Answers | Q 14 | पृष्ठ ४१

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Write the Condition for the Lines → R = → a 1 + λ → B 1 and → R = → a 2 + μ → B 2 to Be Intersecting. - Mathematics

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