Question

Statement I Three vectors $a,b$, and $c$ are coplanar if and only if $a\cdot (b\times c)=0$.
Statement II If $b$ and $c$ are parallel vectors, then $b\times c=a$

• A. Statement I is true; statement II is true; statement II is a correct explanation for statement I
• B. Statement I is true; statement II is true; statement II is not a correct explanation for statement I
• C. Statement I is true; statement II is false
• D. Statement I is false; statement II is true

Answer 1

Answer: (C)

The three vectors $a,b$ and $c$ are coplanar if and only if $a\cdot (b\times c)=0$.
Consider that the vectors $a,b$ and $c$ are coplanar. If $b$ and $c$ are parallel vectors, then $b\times c=0$ and so $a\cdot (b\times c)=0$. If $b$ and $c$ are not parallel then, since $a,b$ and $c$ are coplanar, $b\times c$ is perpendicular to $a$
So, $a\cdot (b\times c)=0$.
Conversely, suppose that a $.(b\times c)=0$. If $a$ and $b\times c$ are both non-zero, then we conclude that $a$ and $b\times c$ are perpendicular vectors. But $b\times c$ is perpendicular to both $b$ and $c$. Therefore, $a,b$ and $c$ must lie in the plane, i.e., they are coplanar. If $a=0$, then $a$ is coplanar with any two vectors, in particular with $b$ and $c$. If $(b\times c)=0$, then $b$ and $c$ are parallel vectors and so, a, b and $c$ are coplanar since any two vectors always lie in a plane determined by them and a vector which is parallel to any one of it also lies in that plane.