# Choose the correct answer if the value of $\hat i.( \hat j\times \hat k)+ \hat j . (\hat i \times \hat k) +\hat k. (\hat i \times \hat j)$ is

$\begin{array} (A)\; 0 \quad & (B)\; -1 \quad & (C)\;1 \quad & (D)\; 3 \end {array}$

Toolbox:
• $\hat i\times \hat j = \hat k$
• $\hat j \times \hat k=\hat i$
• $\hat k \times \hat i = \hat j$
• $\hat i.\hat i = \hat j.\hat j = \hat k.\hat k = 1$
• $\hat i.\hat j = \hat j.\hat k= \hat k.\hat i = 0$
Step 1:
We know $\hat i\times\hat j=\hat k,\hat j\times\hat k=\hat i,\hat k\times\hat i=\hat j$.
$\hat i.\hat i=1,\hat j.\hat j=1$ and $\hat k.\hat k=1$
Let $\hat i.( \hat j\times \hat k)+ \hat j . (\hat i \times \hat k) +\hat k. (\hat i \times \hat j)$------(1)
Step 2:
$\hat i.(\hat j\times\hat k)=\hat i.(\hat j\times\hat k)$
$\hat j.(\hat i\times\hat k)=-\hat j.(\hat k\times\hat i)$
$\hat k.(\hat i\times\hat j)=\hat k.(\hat i\times\hat j)$
Substitute these values in eq(1) we get
$\hat i.( \hat j\times \hat k)+ \hat j . (\hat i \times \hat k) +\hat k. (\hat i \times \hat j)=\hat i.( \hat j\times \hat k)- \hat j . (\hat k \times \hat i) +\hat k. (\hat i \times \hat j)$
$\qquad\qquad\qquad\qquad\qquad\quad\quad\;\;=\hat i.\hat i-\hat j.\hat j+\hat k.\hat k$
We know that $\hat i.\hat i=1,\hat j.\hat j=1$ and $\hat k.\hat k=1$
$\qquad\qquad\qquad\qquad\qquad\quad\quad\;\;=1-1+1$
$\qquad\qquad\qquad\qquad\qquad\quad\quad\;\;=1$
Hence C is the correct answer.
answered May 26, 2013