Question

# Area of a rectangle having vertices A, B, C and D with position vectors$- \widehat i + \dfrac{1}{2}\widehat j + 4\widehat {k,}\widehat i + \dfrac{1}{2}\widehat j + 4\widehat {k,}\widehat i - \dfrac{1}{2}\widehat j + 4\widehat {k,}$ and $- \widehat i - \dfrac{1}{2}\widehat j + 4\widehat k$ respectively is(A) $\dfrac{1}{2}$(B) $1$(C) $2$(D) $4$

Hint: This is a vector algebra based problem. We have been given with the position vectors of all four vertices of a rectangle ABCD. Here, we may involve coordinate geometry methods to find the distance between points and hence to find the area of the rectangle.

First we will find the coordinates of all four vertices with the help of their position vectors.
For vertex A,
Position vector is $- \widehat i + \dfrac{1}{2}\widehat j + 4\widehat k$. So its coordinate will be $\left( { - 1,\dfrac{1}{2},4} \right)$.
For vertex B,
Position vector is $\widehat i + \dfrac{1}{2}\widehat j + 4\widehat k$. So its coordinate will be $\left( {1,\dfrac{1}{2},4} \right)$.
For vertex C,
Position vector is $\widehat i - \dfrac{1}{2}\widehat j + 4\widehat k$.So its coordinate will be$\left( {1, - \dfrac{1}{2},4} \right)$.
For vertex D,
Position vector is $- \widehat i - \dfrac{1}{2}\widehat j + 4\widehat k$.So its coordinate will be$\left( { - 1, - \dfrac{1}{2},4} \right)$.
Distance formula between points $({x_1},{y_1},{z_1})$ and $({x_2},{y_2},{z_2})$ is $\sqrt {{{({x_1} - {x_2})}^2} + {{({y_1} - {y_2})}^2} + {{({z_1} - {z_2})}^2}}$
Now, we can find the length of sides of the rectangle ABCD , let us suppose AB and BC.
AB = Distance between point A and point B
=$\sqrt {{{(1 - ( - 1))}^2} + {{(\dfrac{1}{2} - \dfrac{1}{2})}^2} + {{(4 - 4)}^2}}$
=$\sqrt {{2^2}}$
=$2$
Similarly,
BC= Distance between point B and point C
=$\sqrt {{{(1 - 1)}^2} + {{(\dfrac{1}{2} - ( - \dfrac{1}{2}))}^2} + {{(4 - 4)}^2}}$
=$\sqrt {{1^2}}$
=$1$
Area of rectangle = length of AB $\times$ length of BC
= $2 \times 1$
=$2$
Thus option D is correct.

Note: Vector is an object which has magnitude and direction. This problem is a good example of a geometry related question where coordinates of the points are playing an important role for the computation of other relevant terms of some given shape. Here we have used a distance formula for finding the length and breadth and hence area of the rectangle.