Question

Assertion :In each of the three planes determined by two of the lines $OA,OB,OC$ ($O$ being the origin), a straight line is drawn through $O$ perpendicular to the third line.
The three lines so determined are coplanar. Reason: $(a\times b)\times c+(b\times c)\times a+(c\times a)\times b=0$, where $OA=a,OB=b$ and $OC=c.$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

The plane containg $OA$ and $OB$ is $r.(a\times b)=0$
Any line in this plane is perpendicular to the vector $a\times b$.
Thus the line in this plane perpendicular to $OC$ is parallel to the vector $(a\times b)\times c$

So the equation of three lines are
$r=t(a\times b)\times c,r=p(b\times c)\times a$ and $r=r(c\times a)\times b$
But $(a\times b)\times c+(b\times c)\times a+(c\times a)\times b=0$
So that three vectors $(a\times b)\times c,(b\times c)\times a,(c\times a)\times b$ are coplanar