Question

If $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{b}\times \overrightarrow{c}\ne 0,$ then show that $\overrightarrow{a}+\overrightarrow{c}=m\overrightarrow{b},$ where m is any scalar.

Answer 1

$$\begin{gathered} \overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{b}\times \overrightarrow{c} \\ \Rightarrow \overrightarrow{a}\times \overrightarrow{b}=-\overrightarrow{c}\times \overrightarrow{b} \\ \Rightarrow \overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{c}\times \overrightarrow{b}=0 \\ \Rightarrow \left(\overrightarrow{a}+\overrightarrow{c}\right)\times \overrightarrow{b}=0(\mathrm{Using}\mathrm{right}\mathrm{distributive}\mathrm{property}) \\ \mathrm{Thus},\overrightarrow{a}+\overrightarrow{c}\text{is parallel to}\overrightarrow{b}\text{.} \\ \iff \overrightarrow{a}+\overrightarrow{c}=m\overrightarrow{b},\text{for some scalar}\mathit{\text{m.}} \end{gathered}$$