If a, b, c are coplanar vectors, then - Vidyadip

MCQ

If $ a, b, c$ are coplanar vectors, then

A
$\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right| = 0$
✓
$\left| {\,\begin{array}{*{20}{c}}a&b&c\\{a\,.\,a}&{a\,.\,b}&{a\,.\,c}\\{b\,.\,a}&{b\,.\,b}&{b\,.\,c}\end{array}\,} \right| = 0$
C
$\left| {\,\begin{array}{*{20}{c}}a&b&c\\{c\,.\,a}&{c\,.\,b}&{c\,.\,c}\\{b\,.\,a}&{b\,.\,c}&{b\,.\,b}\end{array}\,} \right| = 0$
D
$\left| {\,\begin{array}{*{20}{c}}a&b&c\\{a\,.\,b}&{a\,.\,a}&{a\,.\,c}\\{c\,.\,a}&{c\,.\,c}&{c\,.\,b}\end{array}\,} \right| = 0$

✓

Answer

Correct option: B.

$\left| {\,\begin{array}{*{20}{c}}a&b&c\\{a\,.\,a}&{a\,.\,b}&{a\,.\,c}\\{b\,.\,a}&{b\,.\,b}&{b\,.\,c}\end{array}\,} \right| = 0$

b
(b) Since $a,\,\,b$ and $c$ are coplanar, therefore there exists

$(x,\,y,\,z$not all zero) such that

$xa + yb + zc = 0$ .....$(i)$

Multiply be $a$ scalarly, we get

$x(a\,.\,a) + (a\,.\,b) + z(a\,.\,c) = 0$ ......$(ii)$

and $x(a\,.\,b) + y(b\,.\,b) + z(b\,.\,c) = 0$ .....$(iii)$

Eliminating $x,\,y$ and $z$ from $(i),$ $(ii)$ and $(iii),$

we get $\left| {\,\begin{array}{*{20}{c}}a&b&c\\{a\,.\,a}&{a\,.\,b}&{a\,.\,c}\\{a\,.\,b}&{b\,.\,b}&{b\,.\,c}\end{array}\,} \right| = 0$.

Note: Students should remember this question as a formula.

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