Did the two ants walk the same distance?

A good answer might be:

No. In this case, it is clear that walking in a straight line to the final destination is shorter.

Summing Displacements =/= Summing Lengths

Summing displacements (vectors) is not the same as summing their length. In the previous example, the sum of u and v yields a vector that is shorter than the length of u plus the length of v. For now, just observe this by looking at the previous diagram, and remembering that "a straight line is the shortest distance between two points." (Later on this will be discussed using the Pythagorean Formula.)

This fact is called the triangle inequality:

length( u + v ) <= length( u ) + length( v )

Here is a case where the length of the sum is much shorter than the sum of the lengths:

e = ( 5, 4 )T
g = ( -4.9, -3.9 )T
e + g = ( .1, .1 )T

(For clarity g has been moved slightly moved away from where it should be.)


Can you think of a situation where the length of the result is equal to the sum of the length of the two input vectors?