For the parallelogram OPQR shown in the sketch, \[\overline {{\rm{OP}}} = {\rm{a\hat t}} + {\rm{b\hat j}}\] and \[\overline {{\rm{OR}}} = {\rm{c\hat t}} + {\rm{d\hat j}}{\rm{.}}\] The area of the parallelogram is A. ad – bc B. ac + bd C. ad + bc D. ab – cd

The area of a parallelogram is given by the formula $A = bh$, where $b$ is the base and $h$ is the height. In the parallelogram OPQR, the base is $\overline{OP} = a\hat{t} + b\hat{j}$ and the height is $\overline{OR} = c\hat{t} + d\hat{j}$. Therefore, the area of the parallelogram is $A = (a\hat{t} + b\hat{j})(c\hat{t} + d\hat{j}) = ac + bd$.

Option A is incorrect because it does not take into account the height of the parallelogram. Option B is incorrect because it does not take into account the base of the parallelogram. Option C is correct because it takes into account both the base and the height of the parallelogram. Option D is incorrect because it does not take into account the direction of the vectors $\overline{OP}$ and $\overline{OR}$.