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10. vector algebra — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths10. vector algebra1 Mark
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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