Find the area of a rectangle whose vertices A,B,C, and D with position vectors  - \(\hat{i}\) +(1/2) \(\hat{j}\)+ 4\(\hat{k}\),   \(\hat{i}\)+ (1/2)\(\hat{j}\)+ 4\(\hat{k}\),   \(\hat{i}\) - (1/2)\(\hat{j}\)+ 4\(\hat{k}\),and   -\(\hat{i}\) -(1/2)\(\hat{j}\)+ 4\(\hat{k}\) respectively-

2 units

The  position vectors of A, B,C, and D of rectangle ABCD are given by:

 \(\vec{OA}\) = - \(\hat{i}\) +(1/2) \(\hat{j}\)+ 4\(\hat{k}\),  \(\vec{OB}\)=  \(\hat{i}\) + (1/2)\(\hat{j}\)+ 4\(\hat{k}\),  \(\vec{OC}\)=  \(\hat{i}\) - (1/2)\(\hat{j}\)+ 4\(\hat{k}\),and \(\vec{OD}\)=   - \(\hat{i}\) -(1/2)\(\hat{j}\)+ 4\(\hat{k}\) respectively

The adjacent sides  \(\vec{AB}\) and  \(\vec{BC}\)   of the given rectangle are given as:

\(\vec{AB}\)= (1+1)\(\hat{i}\) + {(1/2)-(1/2)}\(\hat{j}\)+ (4-4)\(\hat{k}\) = 2\(\hat{i}\)

\(\vec{BC}\) = (1-1)\(\hat{i}\) + {(-1/2)-(-1/2)}\(\hat{j}\)+ (4-4)\(\hat{k}\)=-\(\hat{j}\)

 \(\vec{AB}\)x \(\vec{BC}\)=-2\(\hat{k}\)

|\(\vec{AB}\)x \(\vec{BC}\)| = 2

Hence the area of the given rectangle is 2 units.