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 $=\sqrt{1+4+64}=\sqrt{69}$