Question

If $\hat{i},\hat{j},\hat{k}$ are unit vectors along the positive direction of $X-,Y-$ and $Z-axes$, then a $FALSE$ statement in the following is

• A. $\sum \hat{i}.(\hat{j}\times \hat{k})=0$
• B. $\sum \hat{i}.(\vec{j}+\vec{k})=\vec{0}$
• C. $\sum \hat{i}\times (\hat{j}+\hat{k})=0$
• D. $\sum \hat{i}\times (\hat{j}\times \hat{k})=\vec{0}$

Answer 1

Answer: (A)

Given, $\hat{i},\hat{j},\hat{k}$ are unit vectors along the positive direction of $x,y$ and $z$ -axes, then
(a) $\Sigma \hat{i}\times (\hat{j}+\hat{k})$
$=\hat{i}\times (\hat{j}+\hat{k})+\hat{j}\times (\hat{k}+\hat{i})+\hat{k}\times (\hat{i}+\hat{j})$
$=\hat{k}-\hat{j}+\hat{i}-\hat{k}+\hat{j}-\hat{i}$
$=0$
(b) $\Sigma \hat{i}\times (\hat{j}\times \hat{k})=\hat{i}\times (\hat{j}\times \hat{k})+\hat{j}\times (\hat{k}\times \hat{i})+\hat{k}\times (\hat{i}\times \hat{j})$
$=(\hat{i}\times \hat{i})+(\hat{j}\times \hat{j})+(\hat{k}\times \hat{k})$
$=0+0+0$
$=0$
(c) $\Sigma \hat{i}\cdot (\hat{j}\times \hat{k})=\Sigma (\hat{i}\cdot \hat{i})=\Sigma (1)$
$=1+1+1=3$
(d) $\Sigma \hat{i}\cdot (\hat{j}+\hat{k})=\Sigma (\hat{i}\cdot \hat{j}+\hat{i}\cdot \hat{k})$ $=\Sigma (0+0)=0$