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5Question: A drone follows a straight flight path defined by the parametric equations $ x = 2t + 1, y = -3t + 4, z = t $, where $ t $ is time in seconds. Find the point on this path that is closest to the weather station located at $ (5, -2, 3) $.
To find the point on the line closest to a fixed point, we minimize the square of the distance between a general point on the line and the fixed point. Let the point on the line be $ (2t + 1, -3t + 4, t) $, and the fixed point be $ (5, -2, 3) $. The square of the distance is:
To minimize $ D(t) $, take derivative and set to zero:
ight) $ where $ C(t) $ reaches its minimum.
ight) $), so the minimum occurs at the left endpoint:
ight) $, so the minimum occurs as $ t o
To find the exact value, approach from the right:
ight) $, but the question asks for the value of $ t $ where this occurs. Since $ t =
However, the question asks for the value of $ t $ where the minimum is attained. Since the function is continuous and strictly increasing on the interval, the minimum is attained at the left endpoint not included, so the minimum is not attained.
