The position vectors of the vertices of a quadrilateral ABCD are \[a,\,b,\,c\] and d respectively. Area of the quadrilateral formed by joining the middle points of its sides is [Roorkee 2000]

\[\frac{\text{1}}{\text{4}}\text{ }\!\!|\!\!\text{ b }\!\!\times\!\!\text{ c+c }\!\!\times\!\!\text{ d+d }\!\!\times\!\!\text{ b }\!\!|\!\!\text{ }\].

\[\frac{1}{4}\,|a\times b+b\times d+d\times a|\]

\[\frac{1}{4}\,\left| b\times c+c\times d+a\times d+b\times a \right|\]

\[\frac{1}{4}\,\left| a\times b+b\times c+c\times d+d\times a \right|\]

The position vectors of the vertices of a quadrilateral ABCD are \[a,\

1.	The position vectors of the vertices of a quadrilateral ABCD are \[a,\,b,\,c\] and d respectively. Area of the quadrilateral formed by joining the middle points of its sides is [Roorkee 2000]
A.	\[\frac{1}{4}\,\|a\times b+b\times d+d\times a\|\]
B.	\[\frac{1}{4}\,\left\| b\times c+c\times d+a\times d+b\times a \right\|\]
C.	\[\frac{1}{4}\,\left\| a\times b+b\times c+c\times d+d\times a \right\|\]
D.	\[\frac{\text{1}}{\text{4}}\text{ }\!\!\|\!\!\text{ b }\!\!\times\!\!\text{ c+c }\!\!\times\!\!\text{ d+d }\!\!\times\!\!\text{ b }\!\!\|\!\!\text{ }\].
Answer» D. \[\frac{\text{1}}{\text{4}}\text{ }\!\!\|\!\!\text{ b }\!\!\times\!\!\text{ c+c }\!\!\times\!\!\text{ d+d }\!\!\times\!\!\text{ b }\!\!\|\!\!\text{ }\].

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The position vectors of the vertices of a quadrilateral ABCD are \[a,\,b,\,c\] and d respectively. Area of the quadrilateral formed by joining the middle points of its sides is [Roorkee 2000]

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