Question

If $\overrightarrow{a,}\overrightarrow{b}$are non-collinear vectors, then find the value of $\left[\overrightarrow{a}\overrightarrow{b}\hat{i}\right]\hat{i}+\left[\overrightarrow{a}\overrightarrow{b}\hat{j}\right]\hat{j}+\left[\overrightarrow{a}\overrightarrow{b}\hat{k}\right]\hat{k}.$

Answer 1

$$\begin{gathered} \mathrm{For}\mathrm{any}\mathrm{vector}\overrightarrow{r},\mathrm{we}\mathrm{have} \\ \left(\overrightarrow{r}·\hat{i}\right)\hat{i}+\left(\overrightarrow{r}\mathit{·}\hat{j}\right)\hat{j}+\left(\overrightarrow{r}\mathit{·}\hat{k}\right)\hat{k}=\overrightarrow{r} \\ \mathrm{Replacing}\overrightarrow{r}\mathrm{by}\overrightarrow{a}\times \overrightarrow{b},\mathrm{we}\mathrm{have} \\ \left[\left(\overrightarrow{a}\times \overrightarrow{b}\right)·\hat{i}\right]\hat{i}+\left[\left(\overrightarrow{a}\times \overrightarrow{b}\right)·\hat{j}\right]\hat{j}+\left[\left(\overrightarrow{a}\times \overrightarrow{b}\right)·\hat{k}\right]\hat{k}=\overrightarrow{a}\times \overrightarrow{b} \\ \therefore \left[\overrightarrow{a}\mathit{}\overrightarrow{b}\mathit{}\hat{i}\right]\hat{i}+\left[\overrightarrow{a}\overrightarrow{b}\hat{j}\right]\hat{j}+\left[\overrightarrow{a}\mathit{}\overrightarrow{b}\mathit{}\hat{k}\right]\hat{k}=\overrightarrow{a}\times \overrightarrow{b} \end{gathered}$$