Question

The value of $\hat{i}\times \left(\hat{j}+\hat{k}\right)+\hat{j}\times (\hat{k}+\hat{i})+\hat{k}\times \left(\hat{i}+\hat{j}\right)$ is?

• A. $0$
• B. $\hat{i}$
• C. $\hat{j}$
• D. $\hat{k}$

Answer 1

The correct option is A

$0$

Explanation of the correct option:

Find the value of the given expression.

In the question, an expression $\hat{i}\times \left(\hat{j}+\hat{k}\right)+\hat{j}\times (\hat{k}+\hat{i})+\hat{k}\times \left(\hat{i}+\hat{j}\right)$ is given.

We know that, $\hat{i}\times \hat{j}=\hat{k},\hat{i}\times \hat{k}=-\hat{j},\hat{j}\times \hat{k}=\hat{i},\hat{j}\times \hat{i}=-\hat{k},\hat{k}\times \hat{i}=\hat{j}$ and $\hat{k}\times \hat{j}=-\hat{i}$.

Simplify the given expression as follows:

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

Therefore, the value of the given expression is $0$.

Hence, option $\left(A\right)$ is the correct answer.