274 CHAPTER 13. ALGEBRA AND GEOMETRY OF Rp

illustrated in the following picture in which v+u is illustrated. Note the parallelogramdetermined in the picture by the vectors u and v.

u

v u+v

u

x1

x3

x2

Thus the geometric significance of (d,e, f )+(a,b,c) = (d +a,e+b, f + c) is this. Youstart with the position vector of the point (d,e, f ) and at its point, you place the vectordetermined by (a,b,c) with its tail at (d,e, f ) . Then the point of this last vector will be(d +a,e+b, f + c) . This is the geometric significance of vector addition. Also, as shownin the picture, u + v is the directed diagonal of the parallelogram determined by the twovectors u and v.

The following example is art.

Example 13.3.1 Here is a picture of two vectors u and v.

u

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 thetail of v as shown. Then u+v is the vector which results which is drawn in the followingpretty picture.

uv

u+v

Next consider u−v. This means u+(−v) . From the above geometric description ofvector addition, −v is the vector which has the same length but which points in the oppositedirection to v. Here is a picture.

u

−v

u+(−v)

Finally consider the vector u+2v. Here is a picture of this one also.