Question

If $\left[3\overrightarrow{a}+7\overrightarrow{b}\overrightarrow{c}\overrightarrow{d}\right]=\mathrm{\lambda }\left[\overrightarrow{a}\overrightarrow{c}\overrightarrow{d}\right]+\mathrm{\mu }\left[\overrightarrow{b}\overrightarrow{c}\overrightarrow{d}\right],$ then find the value of λ + μ.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have} \\ \left[3\overrightarrow{a}+7\overrightarrow{b}\mathit{}\overrightarrow{c}\mathit{}\overrightarrow{d}\right]=\lambda \mathit{}\left[\overrightarrow{a}\mathit{}\overrightarrow{c}\mathit{}\overrightarrow{d}\right]+\mu \left[\overrightarrow{b}\mathit{}\overrightarrow{c}\mathit{}\overrightarrow{d}\right] \\ \Rightarrow \left[\left(3\overrightarrow{a}+7\overrightarrow{b}\right)\mathit{\times }\overrightarrow{c}\right]\mathit{.}\overrightarrow{d}=\lambda \mathit{}\left[\overrightarrow{a}\mathit{}\overrightarrow{c}\mathit{}\overrightarrow{d}\right]+\mu \left[\overrightarrow{b}\mathit{}\overrightarrow{c}\mathit{}\overrightarrow{d}\right]\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\left(\mathrm{By}\mathrm{definition}\mathrm{of}\mathrm{scalar}\mathrm{triple}\mathrm{product}\right) \\ \Rightarrow \left[\left(\mathit{3}\overrightarrow{a}\mathit{\times }\overrightarrow{c}\right)+\left(\mathit{7}\overrightarrow{b}\mathit{\times }\overrightarrow{c}\right)\right]\mathit{}\mathit{.}\overrightarrow{d}=\lambda \mathit{}\left[\overrightarrow{a}\mathit{}\overrightarrow{c}\mathit{}\overrightarrow{d}\right]+\mu \left[\overrightarrow{b}\mathit{}\overrightarrow{c}\mathit{}\overrightarrow{d}\right]\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{} \\ \Rightarrow \left(3\overrightarrow{a}\mathit{\times }\overrightarrow{c}\right)\mathit{.}\overrightarrow{d}+\left(7\overrightarrow{b}\mathit{\times }\overrightarrow{c}\right)\mathit{.}\overrightarrow{d}=\lambda \mathit{}\left[\overrightarrow{a}\mathit{}\overrightarrow{c}\mathit{}\overrightarrow{d}\right]+\mu \left[\overrightarrow{b}\mathit{}\overrightarrow{c}\mathit{}\overrightarrow{d}\right]\mathit{}\mathit{}\mathit{}\mathit{} \\ \Rightarrow \left[3\overrightarrow{a}\mathit{}\overrightarrow{c}\mathit{}\overrightarrow{d}\right]+\mathit{}\left[7\overrightarrow{b}\mathit{}\overrightarrow{c}\mathit{}\overrightarrow{d}\right]=\lambda \mathit{}\left[\overrightarrow{a}\mathit{}\overrightarrow{c}\mathit{}\overrightarrow{d}\right]+\mu \left[\overrightarrow{b}\mathit{}\overrightarrow{c}\mathit{}\overrightarrow{d}\right] \\ \Rightarrow 3\mathit{}\left[\overrightarrow{a}\mathit{}\overrightarrow{c}\mathit{}\overrightarrow{d}\right]+7\left[\overrightarrow{b}\mathit{}\overrightarrow{c}\mathit{}\overrightarrow{d}\right]=\lambda \mathit{}\left[\overrightarrow{a}\mathit{}\overrightarrow{c}\mathit{}\overrightarrow{d}\right]+\mu \left[\overrightarrow{b}\mathit{}\overrightarrow{c}\mathit{}\overrightarrow{d}\right]\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\left(\because \mathit{}\left[\lambda \overrightarrow{a}\mathit{}\overrightarrow{b}\mathit{}\overrightarrow{c}\right]\mathit{=}\lambda \left[\overrightarrow{a}\mathit{}\overrightarrow{b}\mathit{}\overrightarrow{c}\right]\mathrm{for}\mathrm{any}\mathrm{scalar}\mathrm{\lambda }\right) \\ \mathrm{Comparing}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \mathrm{\lambda }\mathit{=}3\mathit{} \\ \mu =7 \\ \therefore \mathrm{\lambda }+\mu =3+7=10 \end{gathered}$$