Question

The equation of plane containing the lines $\overrightarrow{\gamma }=\overrightarrow{\alpha }+\lambda \overrightarrow{\beta }$ and $\overrightarrow{\gamma }=\overrightarrow{\beta }+\mu \overrightarrow{\alpha }$

• A. $\left[\begin{array}{ccc}\overrightarrow{\gamma }& \overrightarrow{\alpha }& \overrightarrow{\beta }\end{array}\right]=\overrightarrow{\alpha }.\overrightarrow{\beta }$
• B. $\left[\begin{array}{ccc}\overrightarrow{\gamma }& \overrightarrow{\alpha }& \overrightarrow{\beta }\end{array}\right]=0$
• C. $\left[\begin{array}{ccc}\overrightarrow{\alpha }& \overrightarrow{\beta }& \overrightarrow{\gamma }\end{array}\right]=\overrightarrow{\alpha }\cdot \overrightarrow{\beta }$
• D. None of these

Answer 1

Answer: (B)

$\left[\begin{array}{ccc}\overrightarrow{\gamma }& \overrightarrow{\alpha }& \overrightarrow{\beta }\end{array}\right]=0$

Given lines

$\overrightarrow{\gamma }=\overrightarrow{\alpha }+\lambda \overrightarrow{\beta }$----------------(1)

$\overrightarrow{\gamma }=\overrightarrow{\beta }+\mu \overrightarrow{\alpha }$---------------(2)

planes contain lines i.e it is parallel to line

$\overrightarrow{a}\cdot \overrightarrow{n}=0$

eq of plane will be

$\overrightarrow{\gamma }.\overrightarrow{d}=0$

$\overrightarrow{d}=\overrightarrow{\alpha \times \overrightarrow{\beta }}$

SO

$\overrightarrow{\gamma }.(\overrightarrow{\alpha }\times \overrightarrow{\beta })=0$

$\left[\overrightarrow{\gamma }\overrightarrow{\alpha }\overrightarrow{\beta }\right]=0$