The equation of the plane containing the lines r = a_1 + a_2 and …

MCQ

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

✓
$[r\,\;{a_1}\;\,{a_2}] = 0$
B
$[r\;\,{a_1}\;\,{a_2}] = {a_1}.\;{a_2}$
C
$[r\;\,{a_2}\;\,{a_1}] = {a_1}.\;{a_2}$
D
None of these

✓

Answer

Correct option: A.

$[r\,\;{a_1}\;\,{a_2}] = 0$

a
(a) The required plane passes through a point having position vector ${a_1}$ and is parallel to the vectors ${a_1}$ and ${a_2}$. If $r$ is the position vector of any point on the plane, then $r - {a_1},{a_1},{a_2}$ are coplanar.

Therefore, $(r - {a_1}).({a_1} \times {a_2}) = 0$

==> $[r\,\,{a_1}\,{a_2}]$=$[{a_1}\,{a_1}\,{a_2}] \Rightarrow [r\,{a_1}\,{a_2}] = 0$

Hence, the required plane is $[r\,\,{a_1}\,{a_2}] = 0$.

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