Question

If $ 2\hat{i}+4\hat{j}-5\hat{k} $ and $ \hat{i}+2\hat{j}+3\hat{k} $ are adjacent sides of a parallelogram, then the lengths of its diagonals are

• A. $ 7,\,\sqrt{69} $
• B. $ 6,\,\sqrt{59} $
• C. $ 5,\,\sqrt{65} $
• D. $ 5,\,\sqrt{55} $

Answer 1

Answer: (A)

Let $ \vec{a}=2\hat{i}+4\hat{j}-5\hat{k},\,\,\vec{b}=\hat{i}+2\hat{j}+3\hat{k} $
First diagonal $ =\vec{a}+\vec{b}=3\hat{i}+6\hat{j}-2\hat{k} $
Second diagonal $ =\vec{a}-\vec{b}=\hat{i}+2\hat{j}-8\hat{k} $
Length of first diagonal $ =\sqrt{9+36+4}=\sqrt{49}=7 $
Length of second diagonal $ =\sqrt{1+4+64}=\sqrt{69} $