Question

The two adjacent sides of a parallelogram, $\overrightarrow{a}=2i+4j-5k$ and $\overrightarrow{b}=i+2j+3k$, then find the unit vector parallel to its diagonal.

Answer 1

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ are adjacent side of a parallelogram, where

$\overrightarrow{a}=2\hat{i}+4\hat{j}-5\hat{k}$

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

Let diagonal be $\overrightarrow{c}$

Hence $\overrightarrow{c}=\overrightarrow{a}+\overrightarrow{b}$

$=2\hat{i}+4\hat{j}-5\hat{k}+\hat{i}+2\hat{j}+3\hat{k}=3\hat{i}+6\hat{j}-2\hat{k}$

magnitude of $\overrightarrow{c}=\sqrt{3^2+6^2+2^2}=\sqrt{9+36+4}=7$

Unit vector in direction of $\overrightarrow{c}=\frac{\overrightarrow{c}}{\left|\overrightarrow{c}\right|}=\frac{1}{7}(3\hat{i}+6\hat{j}-2\hat{k})$