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) $\Sigma \hat{ i } \cdot(\hat{ j }+\hat{ k })=\Sigma(\hat{ i } \cdot \hat{ j }+\hat{ i } \cdot \hat{ k })$ $=\Sigma(0+0)=0$