Question

The area of the parallelogram is represented by the vectors $\overrightarrow{A}=2\hat{i}+3\hat{j}$ and $\overrightarrow{B}=\hat{i}+4\hat{j}$

• A. $14\text{units}$
• B. $7.5\mathrm{units}$
• C. $10\mathrm{units}$
• D. $5\mathrm{units}$

Answer 1

The correct option is D

$5\mathrm{units}$

Explanation for Correct Answer:

Determining the area of the parallelogram.

Two given vectors are

$\overrightarrow{A}=2\hat{i}+3\hat{j}$

$\overrightarrow{B}=\hat{i}+4\hat{j}$

These two vectors represent the adjacent sides of a parallelogram

Now, the area of a parallelogram is given by the cross product of two adjacent vector

$\overrightarrow{A}\times \overrightarrow{B}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 3& 0\\ 1& 4& 0\end{array}\right|$

$\begin{array}{l}⟹\overrightarrow{A}\times \overrightarrow{B}=\hat{i}(0-0)-\hat{j}(0-0)+\hat{k}(8-3)\\ =5\hat{k}\\ ⟹|\overrightarrow{A}\times \overrightarrow{B}|=\sqrt{(5{)}^2}\\ =\sqrt{25}\\ =5\end{array}$

Therefore the area of a given parallelogram represented by $\overrightarrow{A}$ and $\overrightarrow{B}$ is equal to $5\mathrm{units}$

Hence, option (D) is correct.