Question

The sides of parallelogram are $2\hat{i}+4\hat{j}-5\hat{k}$ and $\hat{i}+2\hat{j}+3\hat{k}$ Find the unit vectors parallel to their diagonals

• A. $\frac{3}{7}\hat{i}+\frac{6}{7}\hat{j}-\frac{2}{7}\hat{k}$
• B. $\frac{1}{\sqrt{6}9}\hat{i}+\frac{2}{\sqrt{6}9}\hat{j}+\frac{8}{\sqrt{6}9}\hat{k}$
• C. $\frac{3}{7}\hat{i}-\frac{6}{7}\hat{j}+\frac{2}{7}\hat{k}$
• D. $\frac{-1}{\sqrt{6}9}\hat{i}-\frac{2}{\sqrt{6}9}\hat{j}+\frac{8}{\sqrt{6}9}\hat{k}$

Answer 1

Answer: (A), (D)

The correct options are
A $\frac{3}{7}\hat{i}+\frac{6}{7}\hat{j}-\frac{2}{7}\hat{k}$
D $\frac{-1}{\sqrt{6}9}\hat{i}-\frac{2}{\sqrt{6}9}\hat{j}+\frac{8}{\sqrt{6}9}\hat{k}$

Let $\overrightarrow{a}=2\hat{i}+4\hat{j}-5\hat{k}$ and $\overrightarrow{b}=\hat{i}+2\hat{j}+3\hat{k}$ are two sides of a paralleogram.
Diagonals are $\overrightarrow{a}+\overrightarrow{b}=3\hat{i}+6\hat{j}-2\hat{k}$ and $\overrightarrow{b}-\overrightarrow{a}=-\hat{i}-2\hat{j}+8\hat{k}$
$\therefore$ Unit vectors along diagonals are $\frac{3}{7}\hat{i}+\frac{6}{7}\hat{j}-\frac{2}{7}\hat{k}$ and
$\frac{-1}{\sqrt{6}9}\hat{i}-\frac{2}{\sqrt{6}9}\hat{j}+\frac{8}{\sqrt{6}9}\hat{k}$
Hence, options A and D.