question_answer

Find the area of the parallelogram having adjacent sides \[\vec{a}\] and \[\vec{b}\] given by \[2\hat{i}+\hat{j}+\hat{k}\] and \[3\hat{i}+\hat{j}+4\hat{k},\] respectively.

Answer:

Given, \[\overrightarrow{a}=2\,\hat{i}+\hat{j}+\hat{k}\]and \[\overrightarrow{b}=3\,\hat{i}+\hat{j}+4\hat{k}\] Then, vector area of the parallelogram is given by             \[=\hat{i}\,(4-1)-\hat{j}\,(8-3)+\hat{k}\,(2-3)=3\,\hat{i}-5\hat{j}-\hat{k}\]             \[\therefore \]Required area of parallelogram                         \[=\,\,|\overrightarrow{a}\times \overrightarrow{b}|\,\,=\sqrt{{{(3)}^{2}}+{{(-\,5)}^{2}}+{{(-\,1)}^{2}}}\]                         \[=\sqrt{9+25+1}=\sqrt{35}\,\,sq\,\,units\]