108指考數學乙非選第11題詳解

（1）因為 $\overset{\large\rightharpoonup}{DC} = \overset{\large\rightharpoonup}{OC} - \overset{\large\rightharpoonup}{OD}$，且已知 $\overset{\large\rightharpoonup}{OC} = 3\overset{\large\rightharpoonup}{OA}$、$\overset{\large\rightharpoonup}{OD} = 3\overset{\large\rightharpoonup}{OB}$，所以 $$\overset{\large\rightharpoonup}{DC} = 3\overset{\large\rightharpoonup}{OA} - 3\overset{\large\rightharpoonup}{OB} = 3(\overset{\large\rightharpoonup}{OA} - \overset{\large\rightharpoonup}{OB}) = 3\overset{\large\rightharpoonup}{BA}$$ 因為 $\overset{\large\rightharpoonup}{BA} = -\overset{\large\rightharpoonup}{AB} = -(3,-4) = (-3,4)$，所以 $$\overset{\large\rightharpoonup}{DC} = 3(-3,4) = (-9,12)$$ 故 $\overset{\large\rightharpoonup}{DC} = (-9,12)$。 （2）因為 $\overset{\large\rightharpoonup}{AB} = \overset{\large\rightharpoonup}{OB} - \overset{\large\rightharpoonup}{OA} = (3,-4)$，且 $\overset{\large\rightharpoonup}{OA} = (1,2)$，所以 $$\overset{\large\rightharpoonup}{OB} = \overset{\large\rightharpoonup}{OA} + \overset{\large\rightharpoonup}{AB} = (1,2) + (3,-4) = (4,-2)$$ 因為 $\overset{\large\rightharpoonup}{OC} = 3\overset{\large\rightharpoonup}{OA} = 3(1,2) = (3,6)$，且 $\overset{\large\rightharpoonup}{OD} = 3\overset{\large\rightharpoonup}{OB} = 3(4,-2) = (12,-6)$，所以 $\triangle OCD$ 的面積為： $$\frac{1}{2} \left| \det\begin{bmatrix} 3 & 6 \\ 12 & -6 \end{bmatrix} \right| = \frac{1}{2} |3 \times (-6) - 6 \times 12| = \frac{1}{2} |{-18} - 72| = \frac{1}{2} \times 90 = 45$$ 故 $\triangle OCD$ 的面積為 $45$。