Question

If $\hat{i}+\hat{j}-\hat{k}$ and $2\hat{i}-3\hat{j}+\hat{k}$ are adjacent sides of a parallelogram, then the lengths of its diagonals are

• A. $\sqrt{3},\sqrt{14}$
• B. $\sqrt{13},\sqrt{14}$
• C. $\sqrt{21},\sqrt{3}$
• D. $\sqrt{13},\sqrt{21}$

Answer 1

The correct option is D $\sqrt{13},\sqrt{21}$

$\hat{i}+\hat{j}-\hat{k}$ and $2\hat{i}-3\hat{j}+\hat{k}$ are the adjacent sides of a parallelogram.

$\Rightarrow$ the diagonals of the parallelogram can be written as $(\hat{i}+\hat{j}-\hat{k}+2\hat{i}-3\hat{j}+\hat{k})$ and $(\hat{i}+\hat{j}-\hat{k}-(2\hat{i}-3\hat{j}+\hat{k}\left)\right)$

i.e. $(3\hat{i}-2\hat{j})$ and $(-\hat{i}+4\hat{j}-2\hat{k})$

$\therefore$ the lengths of the diagonals would be $\sqrt{3^2+2^2}$ and $\sqrt{(-1{)}^2+4^2+(-2{)}^2}$

i.e. $\sqrt{13},\sqrt{21}$