If the given vectors ( - bc, , b^2 + bc, , c^2 + bc), ( a^2 + ac,… - Vidyadip

MCQ

If the given vectors $( - bc,\,{b^2} + bc,\,{c^2} + bc),$ $({a^2} + ac,\, - ac,\,{c^2} + ac)$ and $({a^2} + ab,\,{b^2} + ab,\, - ab)$ are coplanar, where none of $a, b $ and $ c $ is zero, then

A
${a^2} + {b^2} + {c^2} = 1$
✓
$bc + ca + ab = 0$
C
$a + b + c = 0$
D
${a^2} + {b^2} + {c^2} = bc + ca + ab$

✓

Answer

Correct option: B.

$bc + ca + ab = 0$

b
(b) Accordingly, $\left| {\,\begin{array}{*{20}{c}}{ - bc}&{{b^2} + bc}&{{c^2} + bc}\\{{a^2} + ac}&{ - ac}&{{c^2} + ac}\\{{a^2} + ab}&{{b^2} + ab}&{ - ab}\end{array}\,} \right| = 0$

$ \Rightarrow {(ab + bc + ca)^3} = 0 \Rightarrow ab + bc + ca = 0.$

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