Question

If $\hat{i}+2\hat{j}+3\hat{k},3\hat{i}+2\hat{j}+\hat{k}$ are sides of a parallelogram, then a unit vector parallel to one of the diagonals of the parallelogram is

• A. $\frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}$
• B. $\frac{\hat{i}-\hat{j}+\hat{k}}{\sqrt{3}}$
• C. $\frac{\hat{i}+\hat{j}-\hat{k}}{\sqrt{3}}$
• D. $\frac{-\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}$

Answer 1

The correct option is A $\frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}$

Let $\overrightarrow{a}=\hat{i}+2\hat{j}+3\hat{k}$ and $\overrightarrow{b}=3\hat{i}+2\hat{j}+\hat{k}$.

the two diagonals are $\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}-\overrightarrow{b}$.

So the diagonals are $4(\hat{i}+\hat{j}+\hat{k})$ and $2(-\hat{i}+\hat{k})$.

So the unit vector parallel to one of the diagonal is $\frac{(\hat{i}+\hat{j}+\hat{k})}{\sqrt{3}}$