Question

The adjacent sides of a parallelogram are $a=\hat{i}+2\hat{j}$ and $b=2\hat{i}+\hat{j}$, where $\hat{i}$ and $\hat{j}$ are the usual unit vectors along the positive directions of $x$ and $y$-axes respectively. Then the angle between the diagonals is?

• A. $30°\&150°$
• B. $45°\&135°$
• C. $60°\&120°$
• D. $90°\&90°$

Answer 1

The correct option is D

$90°\&90°$

Explanation for the correct option:

Finding the angle between the diagonals:

The given adjacent sides of a parallelogram,

$a=\hat{i}+2\hat{j}$ and $b=2\hat{i}+\hat{j}$

Considering $\overrightarrow{c}$ as diagonal and $\overrightarrow{d}$ as off-diagonal of the parallelogram then,

$$\begin{gathered} \Rightarrow \overrightarrow{c}=\overrightarrow{a}+\overrightarrow{b} \\ =(\hat{i}+2\hat{j})+(2\hat{i}+\hat{j}) \\ =3\hat{i}+3\hat{j} \end{gathered}$$

And,

$\begin{array}{rcl}& \Rightarrow & \overrightarrow{d}=\overrightarrow{a}-\overrightarrow{b}\\ & =& (\hat{i}+2\hat{j})–(2\hat{i}+\hat{j})\\ & =& -\hat{i}+\hat{j}\end{array}$

Now,

$$\begin{gathered} (\overrightarrow{a}+\overrightarrow{b}).(\overrightarrow{a}–b\overrightarrow{}) \\ =(3\hat{i}+3\hat{j}).(-\hat{i}+\hat{j}) \\ =-3+3\mathbf{[}\mathbf{\because }\hat{\mathbf{i}}\mathbf{,}\mathbf{}\hat{\mathbf{j}}\mathbf{,}\mathbf{}\hat{\mathbf{k}}\mathbf{}\mathbf{\text{are unit vectors}}\mathbf{]} \\ =0 \end{gathered}$$

Since $(\overrightarrow{a}+\overrightarrow{b}).(\overrightarrow{a}–\overrightarrow{b})=0$

So the diagonals are perpendicular.

Thus, the angle between them $=90°$ and $90°$

Hence, option (D) is the correct answer.