Question

if \[\vec{a} \times \vec{b} = \vec{b} \times \vec{c} \neq 0,\]  then  show that \[\vec{a} + \vec{c} = m \vec{b} ,\]  where m is any scalar.

 

 

 

Answer 1

\[\vec{a} \times \vec{b} = \vec{b} \times \vec{c} \]
\[ \Rightarrow \vec{a} \times \vec{b} = - \vec{c} \times \vec{b} \]
\[ \Rightarrow \vec{a} \times \vec{b} + \vec{c} \times \vec{b} = 0\]
\[ \Rightarrow \left( \vec{a} + \vec{c} \right) \times \vec{b} = 0 (\text{ Using right distributive property } )\]
\[\text{ Thus } , \vec{a} + \vec{c} \text{ is parallel to }  \vec{b} .\]
\[ \Leftrightarrow \vec{a} + \vec{c} = m \vec{b} , \text{ for some scalarm } .\]