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\[ - \widehat i + \dfrac{1}{2}\widehat j + 4\widehat {k,}\widehat i + \dfrac{1}{2}\widehat j + 4\widehat {k,}\widehat i - \dfrac{1}{2}\widehat j + 4\widehat {k,}\] and \[ - \widehat i - \dfrac{1}{2}\widehat j + 4\widehat k\] respectively is

(A) $\dfrac{1}{2}$

(B) $1$

(C) $2$

(D) $4$

Answer
Verified

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.