Show that the lines: r = i + j + k + ( i - j + k ) r = i + j + k + (2 i - j + 3 k ) are coplanar. Also, find the equation of the plane containing these lines.

Two lines r = a_ 1 + b_ 1 and r = a_ 2 + b_ 2 are coplanar if = ( a_ 2 + a_ 1 ). ( b_ 1 b_ 2 ). Here (- i + 3 j + k ). [( i - j + k ) (2 i - j + 3 k )] = 0 Equation of plane is ( r - a_ 1 ).( b_ 1 b_ 2 ). = 0 [ r - ( i + j + k ) ]. [( i - j + k ) (2 i - j + 3 k )] = 0 r . (-2 i - j + k ) + 2 = 0

Show that the lines: r = i + j + k + ( i - j + k ) r = i + j + k …

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Question

Show that the lines:
$\overrightarrow{\text{r}} = \hat{\text{i}} + \hat{\text{j}} + \hat{\text{k}} + \lambda(\hat{\text{i}} - \hat{\text{j}} + \hat{\text{k}})$
$\overrightarrow{\text{r}} = \hat{\text{i}} + \hat{\text{j}} + \hat{\text{k}} + \mu(2\hat{\text{i}} - \hat{\text{j}} + 3\hat{\text{k}})$ are coplanar. Also, find the equation of the plane containing these lines.

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Answer

Two lines $ \overrightarrow{\text{r}} = \overrightarrow{\text{a}_{1}} + \lambda \overrightarrow {\text{b}_{1}}$ and $ \overrightarrow{\text{r}} = \overrightarrow{\text{a}_{2}} + \mu \overrightarrow{\text{b}_{2}}$ are coplanar
if $= (\overrightarrow{\text{a}_{2}} + \overrightarrow{\text{a}_{1}}). (\overrightarrow{\text{b}_{1}} \times \overrightarrow{\text{b}_{2}}).$
Here $(-\hat{\text{i}} + 3\hat{\text{j}} + \hat{\text{k}}). [(\hat{\text{i}} - \hat{\text{j}} + \hat{\text{k}})\times(2\hat{\text{i}} - \hat{\text{j}} + 3\hat{\text{k}})] = 0$
Equation of plane is
$(\overrightarrow{\text{r}}- \overrightarrow{\text{a}_{1}}).(\overrightarrow{\text{b}_{1}}\times \overrightarrow{\text{b}_{2}}). = 0$
$\bigg[\overrightarrow{\text{r}} - (\hat{\text{i}} + \hat{\text{j}} + \hat{\text{k}})\bigg]. [(\hat{\text{i}} -\hat{\text{j}} + \hat{\text{k}}) \times(2\hat{\text{i}} - \hat{\text{j}} + 3\hat{\text{k}})] = 0$
$\overrightarrow{\text{r}}. (-2\hat{\text{i}} - \hat{\text{j}} + \hat{\text{k}}) + 2 = 0$