A) \[\vec{n}=(q{{a}_{1}})\hat{i}+(q{{a}_{2}})\hat{j}+(q{{a}_{3}})\hat{k}\]
B) \[\vec{n}=\frac{{{a}_{1}}}{q}\hat{i}+\frac{{{a}_{2}}}{q}\hat{j}+\frac{{{a}_{3}}}{q}\hat{k}\]
C) \[\vec{n}=\frac{q}{{{a}_{1}}}\hat{i}+\frac{q}{{{a}_{2}}}\hat{j}+\frac{q}{{{a}_{3}}}\hat{k}\]
D) \[\vec{n}=\left( \frac{1}{{{a}_{1}}q} \right)\hat{i}+\left( \frac{1}{{{a}_{2}}q} \right)\hat{j}+\left( \frac{1}{{{a}_{3}}q} \right)\hat{k}\]
Correct Answer: C
Solution :
[c] For x-axis \[\vec{r}={{a}_{i}}\hat{i}\,\,\,\,\Rightarrow \,\,\,\,\vec{n}.\hat{j}=\frac{q}{{{a}_{1}}}\] Similarly for y-axis, \[\vec{n}.\hat{j}=\frac{q}{{{a}_{2}}}\] And for z-axis \[\vec{n}.\vec{k}=\frac{q}{{{a}_{3}}}\] \[\therefore \,\,\,\vec{n}=(\vec{n}.\hat{i})\hat{i}+(\vec{n}.\hat{j})\hat{j}+(\vec{n}.\hat{k})\hat{k}\] \[\therefore \,\,\,\vec{n}=\frac{q}{{{a}_{1}}}\hat{i}+\frac{q}{{{a}_{2}}}\hat{j}+\frac{q}{{{a}_{3}}}\hat{k}\]
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