What is a Vector?

Date: 01/04/2002 at 12:25:09
From: Patrick Dornian
Subject: Vectors

Dear Dr. Math,

I am having trouble understanding exactly what a vector is and cannot 
seem to find a simple, straightforward explanation. Please help!

Thank you,
Patrick Dornian

Date: 01/04/2002 at 23:29:18
From: Doctor Peterson
Subject: Re: Vectors

Hi, Patrick.

Vectors can be formally defined in several complicated ways, but I can 
give a basic introduction to the concept.Simply put, a vector is a 
directed quantity.

We'll start with a one-dimensional vector. This is just the same as a 
number. Draw a number line, and draw an arrow starting at zero and 
ending at 5. This is the vector (5). Draw another arrow starting at 
the 5 and ending at the 8; this is the vector (3). It doesn't matter 
where a vector starts; all that matters is how long it is and how far 
it goes. So this second vector is 3 units long and points to the 
right, making it identical to a vector starting at 0 and going to 3. 
By putting two vectors end to end, as I did, I just added the vectors 
(5) and (3) to get the vector (8):

      0   1   2   3   4   5   6   7   8   9
    --+---+---+---+---+---+---+---+---+---+--
      o------------------>o---------->
                5               3
      o------------------------------>
                   5+3=8

If you are familiar with negative numbers, you can see that a vector 
pointing to the left would correspond to a negative number. If we add 
the vectors (5) and (-5), we get the vector (0), a vector with no size 
at all. For any vector you draw, if you move it so that it starts at 
0, it will point to its name.

Since a one-dimensional vector is nothing but a (signed) number, it's 
nothing new. But this introduces the essential concept: only size and 
direction (left or right in this case) count, not position. Now we can 
look at two-dimensional vectors, in a plane, where things start to get 
interesting.

Draw an arrow on a piece of paper, pointing in any direction, and you 
have a vector: a length with a direction. Draw another arrow starting 
at the tip of the first one, and you have added two vectors:

            --+
            / |\
      u+v /     \v
        /        \
      /           \
    o------------->o
            u

If you draw vectors on a coordinate grid, you can give them names:

   V(-2,3) ^          W(3,3)
      +    |         +
       \   |       /  \
       v\  | u+v /     \v
         \ |   /        \
          \| /           \
           o------------->o------>
                   u      U(5,0)

Place each vector so that it starts at the origin (0,0), and name it 
for the point where it ends, just as we did on the number line. Our 
vectors are u = (5,0) and v = (-2,3), since they end at points U 
and V as shown. Their sum w = u+v is (3,3). Do you see how to add two 
vectors? You just add their x coordinates and their y coordinates; 
u goes 5 to the right and v goes 2 to the left, so w goes 5-2 = 3 to 
the right. (Actually, we use the word "coordinate" only for points; 
for vectors, we use the word "component.")

You can do the same for vectors in three-dimensional space, but I 
won't bother drawing that. The important thing is that vectors give us 
a way to talk about anything that has both size and direction, but not 
position - things like velocities, wind speeds, forces, and so on. 
If I row my boat in the direction of vector u, but the water itself 
is moving along vector v, then I will actually be moving along vector 
u+v, so the sum of the two vector velocities tells me how fast, and in 
what direction, I am really going.

For an introduction to vectors with nicer pictures, try this:

   Vectors - Gene Klotz, The Math Forum
   http://mathforum.org/~klotz/Vectors/vectors.html   

Here's another; look under "Basic Stuff":

   Maths Help - Working with vectors - Jenny Olive
   http://www.netcomuk.co.uk/~jenolive/homevec.html   

If you have any further questions, feel free to write back.

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