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

1 Answer
Jun 1, 2018

I get #"40.5 ft"#.

If you want me to round to the nearest integer, it would actually be about #"40 ft"#. Without rounding, it was #40.4598cdots# #"ft"#.

Interestingly enough, the distances each person walked were #(1)# #"20.45 ft"# and #(2)# #"20.02 ft"#, respectively, which add up to be just a LITTLE longer than the total distance between them. So, the angle between the two travel paths must be nearly #180^@#.

(In fact, it is #177.43^@#.)


Well, let's draw this out in 3D space to see what is asked for. I assume up/down refers to altitude (#z# axis). West is #-x# direction, north is #+y# direction, etc.

Suppose person #1# was the #color(blue)("blue")# arrow and person #2# was the #color(red)("red")# arrow. The distance between them is the #color(green)("green")# line.

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 #vecv_1# so that both point in the same direction. That way, the components don't cancel out in the #x# and #z# directions, and we do get a distance farther than #"1 ft"#, which would be physically NOT what is represented on the diagram.

Thus, what we should do is add #-vecv_1# instead:

#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")#