Sum of Two Vectors

Date: 9/10/96 at 16:8:11
From: Anonymous
Subject: Sum of Two Vectors

My daughter is having trouble learning about vectors. How do I explain 
the concept that the sum of two vectors a+b can be equal to or less 
than the sum of vector a + vector b ? To put it another way 
|a + b| <= |a| + |b|. 

I tried to explain it in terms of force being applied in the same 
direction, vs. in slightly different directions. However, no success 
yet. Is there an easier way to express or prove this relation to be 
true ?

Date: 9/10/96 at 18:48:24
From: Doctor Tom
Subject: Re: Sum of Two Vectors

I'd avoid a concept like "force," which may seem vague to her.

Why not this:

I start at home and walk 3 miles in a fixed direction. Then I walk 4 
miles in a fixed direction, but not necessarily in the same direction 
as the first walk. How far from home am I?

It's easy to draw pictures with vectors to see all the possibilities.  
If you happened to continue in exactly the same direction, you'd be 7 
miles away, but it should be clear that every other path will wind up 
closer than 7 miles.  In fact, if you happen to make a 180 degree 
turn, you'll only wind up a mile from home.

-Doctor Tom,  The Math Forum
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