Problem
A guard is located at (-8, 1) and is facing in the direction g = i + 2J. This guard has a field of view of 80°. This means that anything within 40º clockwise or anticlockwise from the direction that the guard is facing can be detected by the guard. For the purpose of this question, the range of sight of this guard is unrestricted. I.e. the distance of an object from a guard does not have any impact whether or not it is detected. (i) A character begins at the location $(6,-5)$ and walks with velocity vector $v=-0.1 i+0.8 j \mathrm{~m} / \mathrm{s}$. At what time will the character first be within the field of view of the guard?
Solution
Step 1 ：Let the guard's position be \((-8, 1)\) and the direction vector be \(g = i + 2j\). The character starts at \((6, -5)\) and moves with a velocity vector \(v = -0.1i + 0.8j\) m/s.
Step 2 ：Calculate the angle between the guard's direction vector and the vector from the guard to the character at each time step.
Step 3 ：When the angle is within 40°, stop the loop and output the time t.
Step 4 ：\(t = 14.32\)
Step 5 ：\(\boxed{t = 14.32 \text{ seconds}}\)
