A) \[\frac{\sqrt{3}}{\sqrt{2}}\]
B) \[\frac{\sqrt{2}}{\sqrt{3}}\]
C) \[\frac{4}{\sqrt{3}}\]
D) \[\frac{2\sqrt{2}}{\sqrt{3}}\]
Correct Answer: A
Solution :
| Normal vector to the plane containing\[\hat{i}+\hat{j}+\hat{k}\] and \[\hat{i}+2\hat{j}+3\hat{k}\] is |
| \[\overrightarrow{n}=\left( \hat{i}+\hat{j}+\hat{k} \right)\times \left( \hat{i}+2\hat{j}+3\hat{k} \right)\] |
| \[\overrightarrow{n}=\hat{i}-2\hat{j}+\hat{k}\] |
| projection of\[(2\hat{i}+3\hat{j}+\hat{k})\]on\[\vec{n}\] |
| \[=\left| \frac{(2\hat{i}+3\hat{j}+\hat{k}).(\hat{i}-2\hat{j}+\hat{k})}{\sqrt{1+4+1}} \right|\] |
| \[=\frac{3}{\sqrt{6}}=\sqrt{\frac{3}{2}}.\] |
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