Magnitudes of Vectors Don't Add Up

Date: 11/14/2010 at 02:15:12
From: David
Subject: Why does the vector law of addition work?

I don't understand how adding vectors results in a triangle in which the
third side is equivalent to the sum of the original two vectors. In
particular, I don't understand how the sum of the two added vectors can
have the same magnitude as the vector sum.

A vector is defined as something with magnitude and direction, so vectors
are equal if and only if they have the same magnitude and direction. The
addition of vectors means combining two vectors, so the result of vector
addition should give a vector with the same direction and magnitude as
that of the combination of the added vectors, right?

I can see how the sum of vectors A and B, if combined, would have the same
direction as the third side of a triangle. What I don't understand here is
how the magnitude of the third side can be equal to the magnitude of the
other two sides. That would mean two sides of a triangle sum to the third
side, wouldn't it?

Date: 11/14/2010 at 10:01:22
From: Doctor Ian
Subject: Re: Why does the vector law of addition work?

Hi David,

Suppose you are standing on a giant grid. You are given two numbers (a,b).
You move a units to the east, and b units to the north. Now you are given
two more numbers (c,d). You move c units to the east, and d units to the
north.

What is the total distance you've moved to the east? It's a + c, right?
And what is the total distance you've moved to the north? It's b + d,
right? So you could have got to the same final point by being given the
numbers (a + c, b + d) along with the same instructions.

Does this make sense? Do you see how it illustrates the rule for vector
addition? 

> A vector is defined as something with magnitude and direction, so
> vectors are equal if and only if they have the same magnitude and
> direction. The addition of vectors means combining two vectors, so the
> result of vector addition should give a vector with the same direction
> and magnitude as that of the combination of the added vectors, right?
 
Right. And they are combined by adding their components, as illustrated in
the example above. 

> I can see how the sum of vectors A and B, if combined, would have the
> same direction as the third side of a triangle. What I don't understand
> here is how the magnitude of the third side can be equal to the
> magnitude of the other two sides. That would mean two sides of a
> triangle sum to the third side, wouldn't it?

The magnitudes don't add directly. If you add two vectors, the magnitude
of the resulting vector will be somewhere between zero and the sum of the
individual magnitudes. 

The latter occurs when they have the same direction, e.g., 

   (3,0) + (4,0) = (7,0)

The vectors on the left have magnitudes of 3 and 4, and the sum has a
magnitude of 7.

The former occurs when they have opposite directions, but the same
magnitude, e.g., 

   (3,0) + (-3,0) = (0,0)

The vectors on the left have magnitude 3, but they cancel each other out,
leaving a null vector, with no direction or magnitude.

In between, we might have something like

   (3,0) + (0,4) = (3,4)

Here, the vectors on the left have magnitudes 3 and 4, but the sum has a
magnitude of 5. That would correspond to a situation like

   Two guys are pushing on a box. One pushes to the east
   with a force of 3 lbs, while the other pushes to the 
   north with a force of 4 lbs. What is the resultant force
   on the box?

We can add the vectors to get (3,4). The magnitude of that is

   sqrt(3^2 + 4^2) = sqrt(25) = 5

The direction of that is

   tan^-1(4/3) = about 53 degrees

So we could replace the two guys with one guy, pushing with a force of 
5 lbs, at an angle of 53 degrees from the x-axis, and the box would move
in the same way as when the two guys push it.

Now, why doesn't the combined force have a magnitude of 7 lbs? Well, think
of it this way: the box is moving at an angle to the force being applied
by the guy pushing to the east. So only SOME of his force is going to
moving the box. And the same is true for the guy pushing to the east. So
we should expect the resultant force to be less than either of the
individual forces.

Now try thinking about those other kinds of cases. In one, the two guys
are both pushing east, and their forces add up -- so the box moves to the
east, under a force of 3 + 4 = 7 lbs. In the other, one guy is pushing
east while the other pushes west, and since they apply the same magnitude
of force, the box doesn't move at all. That is, it's like a force of 
3 + -3 = 0 lbs is being applied to it.

In terms of triangles, you can think of it this way. The hands of a clock
always form two sides of a triangle, right? And the third side of that is
the line connecting the hands. Would you expect the length of that third
side to always be the sum of the lengths of the individual hands? Or does
the angle between them have something to do with it?

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 11/15/2010 at 09:49:43
From: David
Subject: Thank you (Why does the vector law of addition work?)

Yes that clarified it, thank you!!!