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### Plane Flying Parametrically

```
Date: 06/28/99 at 15:58:13
From: Jason
Subject: Plane flying parametrically

I have been trying to figure out how to get the closest result. I
don't know which answer would be correct. The problem is:

The position of an airplane approaching its airport is described
parametrically by P_t = (1000,500,900)+ t[-100,-50,-90]. For what
value of t is the airplane closest to the traffic control center
located at (24,11,13)?
```

```
Date: 06/29/99 at 08:28:39
From: Doctor Jerry
Subject: Re: Plane flying parametrically

Hi Jason,

The x-, y-, z-coordinates of the plane at time t are

x = 1000-100t
y = 500-50t
z = 900-90t.

Often, they are put into vector form, so that the position vector of
the plane at time t is

{x,y,z} = {1000-100t,500-50t,900-90t}

The position vector of the traffic control center is {24,11,13}. You
want the distance from the line to this point. This can be done by
minimizing a function using calculus.

By calculus, minimize

D(t)=((1000-10t)-24)^2+((500-50t)-11)^2+((900-90t)-13)^2.

If you set D'(t)=0 and solve for t you will find t=10.6579, approx.

- Doctor Jerry, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Calculus
High School Physics/Chemistry

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