Find the value of $\hat i\times (\hat j+\hat k)+\hat j\times (\hat k+\hat i)+\hat k\times (\hat i+\hat j)$

Answer 1

Toolbox:

• $\hat i\times \hat i=\hat j\times\hat j=\hat k \times\hat k=0$
• $\hat i\times\hat j=\hat k,\:\:\hat j\times\hat k=\hat i,\:\:\hat k\times\hat i=\hat j$
• $\hat i\times\hat k=-\hat j,\:\:\hat j\times\hat i=-\hat k,\:\:\hat k\times\hat j=-\hat j$

$\hat i\times (\hat j+\hat k)=\hat i\times \hat j+\hat i\times \hat k$

$\qquad=\hat k-\hat j$

Similarly $\hat j\times (\hat k+\hat i)=\hat i-\hat k$

and

$\hat k\times(\hat i+\hat j)=\hat j-\hat i$

Adding all the three we get

$\hat i\times (\hat j+\hat k)+\hat j\times (\hat k+\hat i)+\hat k\times (\hat i+\hat j)$

$=\hat k-\hat j+\hat i-\hat k+\hat j-\hat i=\overrightarrow 0$