Question

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

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

Answer 1

Answer: (B)

(A $\times$B).C is not zero unless B,C are parallel

The correct option is $B$.

It is given that $\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C}=0$

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 $\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C}=0$ will always lie in a single plane forming sides of triangle.

First take

$\overrightarrow{A}\times \overrightarrow{B}=\overrightarrow{B}\times \overrightarrow{C}$

Taking dot product with $C$ on both side of above equation.

$(\overrightarrow{A}\times \overrightarrow{B})\cdot \overrightarrow{C}=(\overrightarrow{B}\times \overrightarrow{C})\cdot \overrightarrow{C}$

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.