[填空题]
A land surveyor is mm2 ya l6;mk,a;eboi2m3)x2lbc3izc go4y b*b85o thxeasuring the area of a triangular plot of land on a smooth hill. She uses a theodolite to determine the coordinates of the corners of the bc3t82gol3 *b oxc ibh xzy)54land and finds them to be (-8,20,10),(16,8,7) and (-20,-8,3) , where the coordinates are in metres from a fixed origin where she stands.
Using vectors, determine the area of the land.A lighthouse positioned at $\mathrm{O}(0,0)$ is located at the centre of a ring shaped coral reef. A sailboat nearby, initially outside the reef, travels at a constant speed of $8 \sqrt{5} \mathrm{~km} \mathrm{~h}^{-1}$ in the direction $\mathbf{d}=\mathbf{i}+2 \mathbf{j}$ . Distances are measured in kilometres.
1. Find the velocity vector of the sailboat, in terms of $ \mathbf{i}$ and $\mathbf{j}$ .
Initially, the sailboat is at the point \mathrm{A}(4,-12) .
2. Find, in terms of $\mathbf{i}$ and $ \mathbf{j}$ , its:
(1) initial position vector;
(2)position vector after t hours, $t \geq 0$ .
3. Find the time when the sailboat is closest to the lighthouse.
The ring-shaped coral reef has a radius of 10 $\mathrm{~km}$ . t = hours
4. Determine whether the sailboat will hit the reef. Justify your answer.
