# Two people meet in the purple room on the fourth floor of a building. On departure, one person travels West 16 feet, South 9 feet, and Down 9 feet. The other person travels North 16 feet, East 8 feet, and Up 9 feet. How far apart are the two people? Round

I get

If you want me to round to the nearest integer, it would actually be about

Interestingly enough, the distances each person walked were

(In fact, it is

Well, let's draw this out in 3D space to see what is asked for. I assume up/down refers to altitude (

Suppose person

The point that they land upon is a vector.

#color(blue)(vecv_1) = << -16, -9, -9 >>#

#color(red)(vecv_2) = << +16, +8, +9 >>#

Do not simply add the vectors together as they are:

#color(blue)(vecv_1) + color(red)(vecv_2) = << 0, -1, 0 >>#

What we actually need to do is *reverse* the vector

Thus, what we should do is add

#color(green)(vecv_3) = -color(blue)(vecv_1) + color(red)(vecv_2)#

#= - << -16, -9, -9 >> + << +16, +8, +9 >>#

#= << +16, +9, +9 >> + << +16, +8, +9 >>#

#= << +32, +17, +18 >>#

Lastly, we need to find the length of this vector to determine the distance:

#color(green)(|vecv_3|) = sqrt(x^2 + y^2 + z^2)#

#= sqrt(("32 ft")^2 + ("17 ft")^2 + ("18 ft")^2)#

#=# #color(green)("40.5 ft")#

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To find the distance between the two people, we can use the Pythagorean theorem in three dimensions. First, we find the displacement vector for each person:

For the first person:

- West: -16 feet
- South: -9 feet
- Down: -9 feet

For the second person:

- North: 16 feet
- East: 8 feet
- Up: 9 feet

Now, we calculate the differences in each direction:

- West to East: ( 16 - (-16) = 32 ) feet
- South to North: ( (-9) - 16 = -25 ) feet
- Down to Up: ( (-9) - 9 = -18 ) feet

Now, we have a displacement vector of ( (32, -25, -18) ).

Using the Pythagorean theorem in three dimensions, the distance between the two people is:

[ \sqrt{(32)^2 + (-25)^2 + (-18)^2} ]

[ = \sqrt{1024 + 625 + 324} ]

[ = \sqrt{1973} ]

Rounded to the nearest foot, the distance between the two people is approximately 44 feet.

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