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If a,b,c are the position vectors of the vertices of a triangle ABC , show that yhe area of the triangle ABC is 1/2 mod( a*b + b*c + c*a ) .

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take a point O as a fixed point 
the vector  a=OA
​                   b=OB
                   c=OC 
AB=OB-OA= b-a

similarly  AC=OC-OA = c-a

                            ​AB*AC=(b-a)*(c-a)
                                          = (b*c)-(b*a)-(a*c)+(a*a)
                                         =(b*c)+(a*b)+(c*a)+0
                                          =​(b*c)+(a*b)+(c*a)
                            => |AB*AC|=|​(b*c)+(a*b)+(c*a)|

Area of triangle =1/2|AB*AC|
                           =1/2|​(b*c)+(a*b)+(c*a)| sq.units . 

 
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