Vector Algebra Question 70
Question: $ \hat{i}\times (\vec{A}\times \hat{i})+\hat{j}\times (\vec{A}\times \hat{j})+\hat{k}\times (\vec{A}\times \hat{k}) $ is equal to
Options:
A) $ \overset{\to }{\mathop{A}}, $
B) $ 2\overset{\to }{\mathop{A}}, $
C) $ 3\overset{\to }{\mathop{A}}, $
D) 0
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Answer:
Correct Answer: B
Solution:
- [b] We have $ \hat{i}\times (\vec{A}\times \hat{i})+\hat{j}\times (\vec{A}\times \hat{j})+\hat{k}\times (\vec{A}\times \hat{k}) $ $ \hat{i}\times (\vec{A}\times \hat{i})=(\hat{i}.\hat{i})\vec{A}-(\hat{i}.\vec{A})\hat{i}=\vec{A}-(\hat{i}.\vec{A})\hat{i} $ ?. (i) $ \hat{j}\times (\vec{A}\times \hat{j})=(\hat{j}.\hat{j})\vec{A}-(\hat{j}.\vec{A})\hat{j}=\vec{A}-(\hat{j}.\vec{A})\hat{j} $ ? (ii) And $ \hat{k}\times (\vec{A}\times \hat{k})=(\hat{k}.\hat{k})\vec{A}-(\hat{k}.\vec{A})\hat{k} $ $ =\vec{A}-(\hat{k}.\vec{A})\hat{k} $ ? (iii) Now, eqn (i) + eqn (ii) + eqn (iii): $ \hat{i}\times (\vec{A}\times \hat{i})+\hat{j}\times (\vec{A}\times \hat{j})+\hat{k}(\vec{A}\times \hat{k})=3\vec{A} $ $ -[(\hat{i}\cdot \vec{A})\hat{i}+(\hat{j}\cdot \vec{A})\hat{j}+(\hat{k}\cdot \vec{A})\hat{k}]=3\vec{A}-\vec{A}=2\vec{A}. $
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