QUESTION2
A character wishes to collect an item that is protected by a security camera. The camera is located at $(2, 2)$ and its detection region is bound by the vectors $c_{1} = - 2 i - 3 j$ and
$c_{2} = 2 i - j$
The item is located at $(3, - 7)$ Once the character collects the item, what direction vector should the character travel along so that it minimises the distance travelled to get beyond the detection zone of the camera? Vector projection should be applied to answer this question.

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Answer

Choose the direction vector that minimizes the distance traveled to get beyond the detection zone: $(4.4, - 2.2)$

Steps

Step 1 ：Let the camera position be $C (2, 2)$ , the item position be $I (3, - 7)$ , and the boundary vectors be $c_{1} = - 2 i - 3 j$ and $c_{2} = 2 i - j$ .

Step 2 ：Find the vector from the camera to the item: $C I = I - C = (1, - 9)$ .

Step 3 ：Find the vector projection of $C I$ onto $c_{1}$ and $c_{2}$ :

Step 4 ： $p r o j_{c_{1}} (C I) = \frac{C I \cdot c_{1}}{c_{1} \cdot c_{1}} c_{1} = \frac{(1, - 9) \cdot (- 2, - 3)}{(- 2, - 3) \cdot (- 2, - 3)} (- 2, - 3) = (- 3.84615385, - 5.76923077)$

Step 5 ： $p r o j_{c_{2}} (C I) = \frac{C I \cdot c_{2}}{c_{2} \cdot c_{2}} c_{2} = \frac{(1, - 9) \cdot (2, - 1)}{(2, - 1) \cdot (2, - 1)} (2, - 1) = (4.4, - 2.2)$

Step 6 ：Choose the direction vector that minimizes the distance traveled to get beyond the detection zone: $(4.4, - 2.2)$

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