Consider the
vectors u = (3, 4) and
v = (4, 1) in the plane. From the
component method of vector
addition we know that the
sum of these two
vectors is
u + v = (7, 5). Graphically, we see that this is the
same as the result we would get by "picking up" one of the vectors (without
changing either its
direction or its
magnitude), placing its end at the
other (unmoved) vector's tip, and drawing an arrow from the origin to the new
tip location for the displaced vector.
This geometric procedure for adding vectors works in general. For any two
vectors
u and
v in the plane, the sum of the vectors is graphically given as
in the following figure:
Figure 2.2: The sum of the vectors u and v in the plane.
The geometric procedure is valid for 3-dimensional vectors as well.
Notice that in the same way that any two lines lie in a plane, any two
vectors in 3-dimensional space will also lie in the same plane. This
recognition allows us to see that the sum of two vectors will always lie in the
plane defined by the original two vectors.
As we noted in
Vector
Subtraction, in order to
subtract one vector from another, you simply add its negative partner:
u - v=
u + (- 1)v. Thus, vectors can be subtracted graphically in the same manner
used for adding them, by simply taking care to reverse the
direction of the
vector being subtracted:
Figure 2.3: The difference of the vectors u and v in the plane.
If you graphically add back in the subtracted vector to your result from the
subtraction and you recover the initial vector you subtracted from. In other
words,
(u - v) + v = u in our graphical methods, as we should expect!
