Question

The adjacent sides of a parallelogram are represented by the vectors $\overrightarrow{a}=\hat{i}+\hat{j}-\hat{k}\mathrm{and}\overrightarrow{b}=-2\hat{i}+\hat{j}+2\hat{k}.$ Find unit vectors parallel to the diagonals of the parallelogram.

Answer 1

$\begin{array}{l}\overrightarrow{a}=\hat{i}+\hat{j}-\hat{k}\\ \overrightarrow{b}=-2\hat{i}+\hat{j}+2\hat{k}\end{array}$

$\begin{array}{ccl}\overrightarrow{AC}& =& \overrightarrow{a}+\overrightarrow{b}\\ & =& -\hat{i}+2\hat{j}+\hat{k}\end{array}$

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{unit}\mathrm{vector}\mathrm{along}\mathrm{the}\mathrm{diagonals}AC\mathrm{and}BD\mathrm{of}\mathrm{the}\mathrm{parallelogram}\mathrm{be}\widehat{AC}\mathrm{and}\widehat{BD}. \\ \Rightarrow \hspace{0.17em}\hspace{0.17em}\widehat{AC}=\frac{-\hat{i}+2\hat{j}+\hat{k}}{\sqrt{6}} \end{gathered}$$

$\begin{array}{ccc}\mathrm{Similarly},\hspace{0.17em}\mathit{\hspace{0.17em}}\overrightarrow{BD}& =& \overrightarrow{b}-\overrightarrow{a}\\ & =& -3\hat{i}+3\hat{k}\end{array}$

$\Rightarrow \hspace{0.17em}\hspace{0.17em}\widehat{BD}=\frac{-3\hat{i}+3\hat{k}}{3\sqrt{2}}=\frac{-\hat{i}+k}{\sqrt{2}}$