Question

Given that A × B=O, B × C=O, and A≠0, B≠0, C≠0. Find the value of A × C

Answer 1

Task. Given that $A\times B=O$, $B\times C=O$, and $A\neq 0$, $B\neq 0$, $C\neq 0$. Find the value of $A\times C$.

Solution. Recall that for any two vectors $A$ and $B$ in $\mathbb{R}^{3}$ the absolute value of their cross product $|A\times B|$ is equal to

$|A\times B|=|A|\cdot|B|\cdot\sin\widehat{AB},$

where $\widehat{AB}$ is the angle between vectors $A$ and $B$.

By assumption $|A|\neq 0$, $|B|\neq 0$, $|C|\neq 0$, and

$|A|\cdot|B|\cdot\sin\widehat{AB}=0,\qquad|B|\cdot|C|\cdot\sin\widehat{CB}=0.$

Therefore

$\sin\widehat{AB}=\sin\widehat{BC}=0.$

This means that vectors $A$ and $B$ are collinear, and similarly $B$ and $C$ are also collinear. Hence, $A$ and $C$ are also collinear, and so

$\sin\widehat{AC}=0.$

Therefore

$|A\times C|=|A|\cdot|C|\cdot\sin\widehat{AC}=0,$

which means that $A\times C=0$.

Answer. $A\times C=0$.