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If \overrightarrow{a}=\widehat{i}+2\widehat{j}+3\widehat{k} and \overrightarrow{b}=2\widehat{i}+4\widehat{j}-5\widehat{k} represent two adjacent sides of a parallelogram, find unit vectors parallel to the diagonals of the parallelogram.

 

Answers (1)

Given; 

\overrightarrow{a}=\widehat{i}+2\widehat{j}+3\widehat{k} 

\overrightarrow{b}=2\widehat{i}+4\widehat{j}-5\widehat{k} 

\\ \text { Diagonal vectors are: } \vec{a}+\vec{b} , \vec{a}-\vec{b}

\\ \quad \vec{a}+\vec{b}=3 \hat{i}+6 \hat{j}-2 \hat{k} \\ \vec{a}-\vec{b}=-\hat{i}-2 \hat{j}+8 \hat{k} \\

\\|\vec{a}+{\vec{b}}| = \sqrt{3^2+6^2+(-2)^2} = 7 \\|\vec{a}-{\vec{b}}| = \sqrt{(-1)^2+(-2)^2+(8)^2} = \sqrt{69} \\

\\ \text {Unit vectors } \\\\ \frac{(\vec{a}+\vec{b})}{|\vec{a}+{\vec{b}}|}=\frac{3}{7} \hat{i}+\frac{6}{7} \hat{j}-\frac{2}{7} \hat{k} \\ \frac{(\vec{a}-\vec{b})}{|\vec{a}-\vec{b}|}=-\frac{1}{\sqrt{69}} \hat{i}-\frac{2}{\sqrt{69}} \hat{j}+\frac{8}{\sqrt{69}} \hat{k}

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