Question: For any vector $ \vec{\alpha } $ , what is $ ( \overset{\to }{\mathop{\alpha }},.\widehat{i} )\widehat{i}+( \overset{\to }{\mathop{\alpha }},.\widehat{j} )\widehat{j}+( \overset{\to }{\mathop{a}},.\widehat{k} )\widehat{k} $ equal to?
Options:
A) $ \overset{\to }{\mathop{\alpha }}, $
B) $ 3\overset{\to }{\mathop{\alpha }}, $
C) $ -\overset{\to }{\mathop{\alpha }}, $
D) $ \overset{\to }{\mathop{0}}, $
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Answer:
Correct Answer: A
Solution:
- [a] Let $ \overrightarrow{\alpha }=\overrightarrow{a},\hat{i}+\overrightarrow{b}\hat{j}+\overrightarrow{c}\hat{k} $ Now, $ \overrightarrow{\alpha },\hat{i}=( \overrightarrow{a},\hat{i}+\overrightarrow{b},\hat{j}+\overrightarrow{c},\hat{k} ).\hat{i}=\overrightarrow{a} $ $ \overrightarrow{\alpha },\hat{j}=( \overrightarrow{a},\hat{i}+\overrightarrow{b},\hat{j}+\overrightarrow{c},\hat{k} ).\hat{j}=\overrightarrow{b} $ $ \overrightarrow{\alpha },\hat{k}=( \overrightarrow{a},\hat{i}+\overrightarrow{b},\hat{j}+\overrightarrow{c},\hat{k} ).\hat{k}=\overrightarrow{c} $ Now, $ \overrightarrow{a},\hat{i}+\overrightarrow{b},\hat{j}+\overrightarrow{c},\hat{k}=\overrightarrow{\alpha } $ Thus, required expression = $ \overrightarrow{\alpha } $ .
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