Question

A parallelogram is formed with $\overrightarrow{a}$ and $\overrightarrow{b}$ as the sides. Let $\overrightarrow{d_1}$ and $\overrightarrow{d_2}$ be the diagonals of the parallelogram, then $a^2+b^2=$

• A. $(d_1^2+d_2^2)/2$
• B. $(d_1^2-d_2^2)/2$
• C. $d_1^2+d_2^2$
• D. $d_1^2-d_2^2$

Answer 1

The correct option is A $(d_1^2+d_2^2)/2$

Let $\theta$ be the angle between $\overrightarrow{a}\&\overrightarrow{b}$
The long diagonal of the parallelogram is the vector sum of two sides and short diagonal is vector difference of two sides.
Given...$\overrightarrow{d_1}=\overrightarrow{a}+\overrightarrow{b},\overrightarrow{d_2}=\overrightarrow{a}-\overrightarrow{b}$
$d_1^2=a^2+b^2+2ab\cos \theta d_2^2=a^2+b^2-2ab\cos \theta$
$d_1^2+d_2^2=2(a^2+b^2)$
$a^2+b^2=(d_1^2+d_2^2)/2$