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. $\left(\vec{A}\times\vec{B}\right)\cdot\vec{C}=\left|\vec{A}\right|\left|\vec{B}\right|\left|\vec{C}\right| \rightarrow C^{2}=A^{2}+B^{2}$

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.