18 CHAPTER 2. Fn

Definition 2.2.1 Let x = (x1, · · · ,xn) be the coordinates of a point in Rn. Imagine an arrow(line segment with a point) with its tail at 0 = (0, · · · ,0) and its point at x as shown in thefollowing picture in the case of R3.

(x1,x2,x3) = x

Then this arrow is called the position vector of the point x. Given two points, P,Qwhose coordinates are (p1, · · · , pn) and (q1, · · · ,qn) respectively, one can also determinethe position vector from P to Q defined as follows.

−→PQ≡ (q1− p1, · · · ,qn− pn)

Thus every point in Rn determines a vector and conversely, every such position vector(arrow) which has its tail at 0 determines a point of Rn, namely the point of Rn whichcoincides with the point of the position vector. Also two different points determine aposition vector going from one to the other as just explained.

Imagine taking the above position vector and moving it around, always keeping it point-ing in the same direction as shown in the following picture. After moving it around, it isregarded

•(x1,x2,x3) = x

as the same vector because it points in the samedirection and has the same length.2Thus each ofthe arrows in the above picture is regarded as thesame vector. The components of this vector are thenumbers, x1, · · · ,xn obtained by placing the initialpoint of an arrow representing the vector at the ori-gin. You should think of these numbers as direc-tions for obtaining such a vector illustrated above.Starting at some point (a1,a2, · · · ,an) in Rn, youmove to the point (a1 + x1, · · · ,an) and from thereto the point (a1 + x1,a2 + x2,a3 · · · ,an) and then to

(a1 + x1,a2 + x2,a3 + x3, · · · ,an) and continue this way until you obtain the point

(a1 + x1,a2 + x2, · · · ,an + xn) .

The arrow having its tail at (a1,a2, · · · ,an) and its point at

(a1 + x1,a2 + x2, · · · ,an + xn)

looks just like (same length and direction) the arrow which has its tail at 0 and its point at(x1, · · · ,xn) so it is regarded as representing the same vector.

2.3 Geometric Meaning Of Vector AdditionIt was explained earlier that an element of Rn is an ordered list of numbers and it wasalso shown that this can be used to determine a point in three dimensional space in the

2I will discuss how to define length later. For now, it is only necessary to observe that the length should bedefined in such a way that it does not change when such motion takes place.