Show that the lines \overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b} and \overrightarrow{r}=\overrightarrow{b}+\mu \overrightarrow{a} are coplanar and the plane containing them is given by \overrightarrow{r} \cdot(\overrightarrow{a} \times \overrightarrow{b})=0.

CBSE Class 12 Math 2020 Outside Delhi Set 1 Solved Paper

Question : 36 of 36

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Show that the lines

{r}↖{→}={a}↖{→}+λ {b}↖{→}

and

{r}↖{→}={b}↖{→}+µ {a}↖{→}

are coplanar and the plane containing them is given by

{r}↖{→} ⋅({a}↖{→} × {b}↖{→})=0

Solution:

Condition for Coplanarity of Two Lines in Vector Form
If the lines

{r}↖{→}={a_1}↖{→}+λ {b_1}↖{→}

and

{r}↖{→}={a_2}↖{→}+µ {b_2}↖{→}

are coplanar,
then

{r}↖{→} ∙ ({b_1}↖{→} × {b_2}↖{→}) ={a_2}↖{→} ∙ ({b_1}↖{→} × {b_2}↖{→})

Substituting above values in this equations

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