Question

Write the value of $\hat{i}\times \left(\hat{j}+\hat{k}\right)+\hat{j}\times \left(\hat{k}+\hat{i}\right)+\hat{k}\times \left(\hat{i}+\hat{j}\right).$

Answer 1

$$\begin{gathered} \hat{i}\times \left(\hat{j}+\hat{k}\right)+\hat{j}\times \left(\hat{k}+\hat{i}\right)+\hat{k}\times \left(\hat{i}+\hat{j}\right) \\ =\left(\hat{i}\times \hat{j}\right)+\left(\hat{i}\times \hat{k}\right)+\left(\hat{j}\times \hat{k}\right)+\left(\hat{j}\times \hat{i}\right)+\left(\hat{k}\times \hat{i}\right)+\left(\hat{k}\times \hat{j}\right) \\ =\hat{k}-\hat{j}+\hat{i}-\hat{k}+\hat{j}-\hat{i} \\ =\overrightarrow{0} \end{gathered}$$