Question

The diagonals of a parallelogram are $2\hat i$ and $2\hat j$. What is the area of the parallelogram?
A). $0.5$ unit
B). $1$ unit
C). $2$ unit
D). $4$ unit

Answer 1

A parallelogram is a two-dimensional, geometric figure with four sides in which the opposite sides are parallel and equal to each other. Diagonals of a parallelogram are the lines that connect the opposite corners of the figure. The area of parallelogram is given by in terms of diagonals is given by $Area = \dfrac{1}{2} \times \left| {{D_1} \times {D_2}} \right|$ where ${D_1}$ and ${D_2}$ are the diagonals of the parallelogram.

Complete step-by-step solution:
The diagonals of a parallelogram are $2\hat i$ and $2\hat j$.
Therefore, ${D_1}$ = $2\hat i$ and ${D_2}$ = $2\hat j$
The area of parallelogram is given by in terms of diagonals is given by $Area = \dfrac{1}{2} \times \left| {{D_1} \times {D_2}} \right|$ where ${D_1}$ and ${D_2}$ are the diagonals of the parallelogram.
Substituting the diagonal values in the formula,
$Area = \dfrac{1}{2} \times 2\hat i \times 2\hat j$
Canceling $2$from numerator and denominator and taking common,
$Area = \left|2(\hat i \times \hat j)\right|$
$\hat i$ and $\hat j$ are the vector components of a three-dimensional vector plane.
If we take the cross product of vector components $\hat i$ and $\hat j$ we get the resultant as $\hat k$ .
i.e., $\hat i \times \hat j = \hat k$
$\therefore Area = \left| 2\hat k \right|$
Hence, the area of a parallelogram is $2$ units with diagonals $2\hat i$ and $2\hat j$ .
The correct option is C. $2$ units.

Note: The vector cross product of components of a three-dimensional vector plane is given as $\hat i \times \hat j = \hat k$ , $\hat j \times \hat k = \hat i$ and $\hat k \times \hat i = \hat j$. The coefficient of the vector component is the scalar value of that component. As in the above example, the $2$ unit is the scalar value of the area of a parallelogram in a three-dimensional vector plane.