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Vectors — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsVectors2 Marks
MCQ
If $\bar{a}, \bar{b}, \bar{c}$ are three non-zero vectors which are pairwise non-collinear. If $\bar{a}+3 \bar{b}$ is collinear with $\bar{c}$ and $\bar{b}+2 \bar{c}$ is collinear with $\bar{a}$, then $\overline{ a }+3 \overline{b}+6 \overline{ c }$ is
  • A
    $\bar{c}$
  • ✓
    $\overline{0}$
  • C
    $\bar{a}+\bar{c}$
  • D
    $\overline{ a }$
✓

Answer

Correct option: B.
$\overline{0}$
(B) Since $\overline{ a }+3 \overline{b}$ is collinear with $\overline{ c }$, and $\overline{ b }+2 \overline{ c }$ is collinear with $\overline{ a }$,
$\therefore \overline{ a }+3 \overline{b}=x\overline{ c }$ and $\overline{ b }+2 \overline{ c }=y\overline{ a } \ \forall x, y \in R$.
$\therefore \bar{a}+3 \bar{b}+6 \bar{c}=(+6) \bar{c}$
Also, $\overline{ a }+3 \overline{b}+6 \overline{ c }=\overline{ a }+3(\overline{b}+2 \overline{ c })=(1+3y) \overline{ a }$
$\therefore (x+6) \bar{c}=(1+3y) \bar{a}$
$\Rightarrow(x+6) \overline{ c }-(1+3 y) \overline{ a }=0$
$\therefore x+6=0$ and $1+3 y=0$
$\Rightarrow x=-6$ and $y=-\frac{1}{3}$
Now, $\overline{ a }+3 \overline{b}=x \overline{ c } \Rightarrow \overline{ a }+3 \overline{b}+6 \overline{ c }=\overline{0}$

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