Q.If in parallelogram ABCD, diagonal vectors are $\overrightarrow{AC}=2\hat i+3\hat j+4\hat k$ and $\overrightarrow{BD}=-6\hat i+7\hat j-2\hat k$, then find the adjacent side vectors $\overrightarrow{AB}$ and $\overrightarrow{AD}$.

In parallelogram $ABCD$: $\overrightarrow{AC}=\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AB} +\overrightarrow{AD}$ (since $BC=AD$ in a parallelogram), and $\overrightarrow{BD}=\overrightarrow{BA} +\overrightarrow{AD}=-\overrightarrow{AB}+\overrightarrow{AD}=\overrightarrow{AD}-\overrightarrow{AB}$.

Given $\overrightarrow{AC}=2\hat i+3\hat j+4\hat k$ and $\overrightarrow{BD}=-6\hat i+7\hat j-2\hat k$:

$\overrightarrow{AC}+\overrightarrow{BD}=2\overrightarrow{AD}\Rightarrow\overrightarrow{AD} =\frac{(2-6)\hat i+(3+7)\hat j+(4-2)\hat k}{2}=\frac{-4\hat i+10\hat j+2\hat k}{2}=-2\hat i+5\hat j+\hat k.$ …

Express the two diagonals AC and BD in terms of the two adjacent sides AB and AD, then add and subtract these t …

Using AB = AC + BD (or similar wrong combination) instead of correctly solving th …