Question

The equation of the plane containing the lines $r=a_1+\lambda b$ and $r=a_2+\mu b$ is

• A. $r.(a_1-a_2)\times b=\left[a_1{a}_{2}b\right]$
• B. $r.(a_2-a_1)\times b=\left[a_1{a}_{2}b\right]$
• C. $r.(a_1+a_2)\times b=\left[a_1{a}_{2}b\right]$
• D. None of these

Answer 1

The correct option is B $r.(a_2-a_1)\times b=\left[a_1{a}_{2}b\right]$

The required plane passes through the points having position vectors $a_1$ and $a_2$ and is parallel to the vector $b$.

Therefore, if $r$ is the position vector of any variable point on the plane, then the vectors $r-a_1,a_2-a_1$ and $b$ are coplanar.

$\therefore (r-a_1).((a_2-a_1)\times b)=0\Rightarrow r.(a_2-a_1)\times b-a_1.(a_2\times b)=0\Rightarrow r.(a_2-a_1)\times b=\left[a_1{a}_{2}b\right]$