Question

Three vectors $\vec{A}$, $\vec{B}$ and $\vec{C}$ add up to zero. Find which is false.

• A. $\left(\vec{A}\times \vec{B}\right)\times \vec{C}$ is not zero unless $\vec{B}$, $\vec{C}$ are parallel
• B. $\left(\vec{A}\times \vec{B}\right)\cdot \vec{C}$ is not zero unless $\vec{B}$, $\vec{C}$ are parallel
• C. If $\vec{A}$, $\vec{B}$, $\vec{C}$ define a plane, $\left(\vec{A}\times \vec{B}\right)\times \vec{C}$ is in that plane
• D. $\left(\vec{A}\times \vec{B}\right)\cdot \vec{C}=\left|\vec{A}\right|\left|\vec{B}\right|\left|\vec{C}\right|\to C^2=A^2+B^2$

Answer 1

Answer: (B)

The correct option is $B$.
It is given that $\vec{A}+\vec{B}+\vec{C}=0$
Now if vector triple product of $A$ and $B$ and $C$, then vector will always lie on the plane
which will be formed by $A,B$ and $C$.
It means $\vec{A}+\vec{B}+\vec{C}=0$
will always lie in a single plane forming sides of triangle.
First take $\vec{A}\times \vec{B}=\vec{B}\times \vec{C}$
Taking dot product with $C$ on both side of above equation.
$(\vec{A}\times \vec{B})\cdot \vec{C}=(\vec{B}\times \vec{C})\cdot \vec{C}$
Now this will be zero on two conditions.
First is that $B$ and $C$ are parallel to each other.
But it could be zero without $C$ being parallel to $B$.
As when we will take the cross product of $B$ and $C$ vectors,
then any vector perpendicular (say $P$ ) to both $B$ and $C$.
Then by taking the dot product of $P$ and $C$ will also be zero as the angle between them will always be $90$ degrees.
So, statement $B$ is false.