Answer
424.5k+ views
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.
Complete step-by-step answer:
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.
Complete step-by-step answer:
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.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)