Question

please show work and explain i dont understand at all

op lxk)*(kxi) is: The result of

Answer 1

The vector product of two vectors is given by

  

i A x B = Ay Ay A. B. By B.

Where

$\vec{A}= A_{x}\, \hat{i}+A_{y}\, \hat{j}+A_{z}\,\hat{k}$

and   

B = B. + By j + Bok

Now, the unit vectors $\hat{i}$ , $\hat{j}$ and $\hat{k}$ can be writtten as

  $\hat{i} = 1. \hat{i}+0. \hat{j} +0. \hat{k}$

$\hat{j} = 0. \hat{i}+1. \hat{j} +0. \hat{k}$  

and $\hat{k} = 0. \hat{i}+0. \hat{j} +1. \hat{k}$

So,

  $\hat{j} \times \hat{k}= \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix}= \hat{i} .(1-0)- \haat{j}.(0-0)+\hat{k}.(0-0)= \hat{i}$

Similarly,

$\hat{k} \times \hat{i}= \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{vmatrix}= \hat{i} .(0-0)- \haat{j}.(0-1)+\hat{k}.(0-0)= \hat{j}$

And,

  $\hat{i} \times \hat{j}= \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{vmatrix}= \hat{i} .(0-0)- \haat{j}.(0-0)+\hat{k}.(1-0)= \hat{k}$

Therefore,

  $(\hat{j} \times \hat{k}) \times (\hat{k} \times \hat{i})= \hat{k}$