13.3 Geometric Meaning Of Vector Addition In ℝ3
It was explained earlier that an element of ℝn is an n tuple of numbers and it was also
shown that this can be used to determine a point in three dimensional space in the case
where n = 3 and in two dimensional space, in the case where n = 2. This point was
specified relative to some coordinate axes.
Consider the case where n = 3 for now. If you draw an arrow from the point in three
dimensional space determined by
to the point
with its tail sitting at the
and its point at the point
this arrow is called the position
of the point determined by u ≡
One way to get to this point is to start
and move in the direction of the
and then in the direction
and finally in the direction of the
evident that the same arrow (vector) would result if you began at the point
moved in the direction of the x1
then in the
direction of the x2
and finally in the x3
only this time, the arrow would have its tail sitting at the point
and its point at
It is said to be the
same arrow (vector) because it will point in the same direction and have the
same length. It is like you took an actual arrow, the sort of thing you shoot
with a bow, and moved it from one location to another keeping it pointing the
same direction. This is illustrated in the following picture in which v
illustrated. Note the parallelogram determined in the picture by the vectors u
Thus the geometric significance of
You start with the position vector of the point
and at its point, you place the
vector determined by
with its tail at
Then the point of this last vector
This is the geometric significance of vector addition. Also, as
shown in the picture, u
is the directed diagonal of the parallelogram determined by
the two vectors u
The following example is art.
Example 13.3.1 Here is a picture of two vectors u and v.
Sketch a picture of u + v,u − v, and u+2v.
First here is a picture of u + v. You first draw u and then at the point of u you place
the tail of v as shown. Then u + v is the vector which results which is drawn in the
following pretty picture.
Next consider u − v. This means u+
From the above geometric description of
vector addition, −v
is the vector which has the same length but which points in the
opposite direction to v
. Here is a picture.
Finally consider the vector u+2v. Here is a picture of this one also.