Another way to look at vectors is to consider them as ordered pairs of numbers.  We don’t use this description much in early physics classes, but it is a useful way to think about vectors.

Vectors as ordered pairs

Picture a displacement vector placed on a coordinate grid as shown.  The vector starts at position A = (4,5) and ends up at position B = (7,7).  A and B are the coordinates of the vector, not the vector itself.

displacement using ordered pairs
The displacement vector tells us how to move to get from A to B.  We need to move a distance (7-4) = 3 in the x-direction, and a distance (7-5) =2 in the  y-direction.  We would write this as

\(\color{black}{\vec{d} = ((7-4),(7-5))}\) or

\(\color{black}{\vec{d} = (3,2)}\)


The vector d could be drawn in other locations.  It is a displacement vector, telling us how to get from one spot to another.  It doesn’t necessarily care where the initial and final spots are:  It shows how far apart two points are, and in which direction one moves.  As long as the size and relative direction don’t change, the vector remains constant, regardless of where we move it.

Notice that we write vector d with a small arrow above it.  This denotes that it is a vector.  The ordered pair notation itself is not clearly a vector, but it turns out that ordered pairs generally are vectors.  If we had drawn an arrow from the origin to point A, it would be a vector too.

\(\color{black}{\vec{A} = ((4-0),(5-0))}\) or

\(\color{black}{\vec{A} = (4,5)}\)

position-and-displacementFrom the picture you can see the B vector would be found similarly.

More advanced Vectors

In later physics classes, we will see vectors described by three sets of numbers.  This occurs when we move from a two dimensional (x and y) space to a three dimensional space (x,y,z).

In advanced math classes, you might see even more sets of numbers.  There are a few extra guidelines that determine whether a set of numbers is a vector or not.  For physics, we will usually work with two dimensions, but might occasionally consider three dimensions.